I would say that the questions that String/M-theory try to answer are the last ones that our current knowledge of reality allows us to ask.
One may think they are the last ones because they are already indirect, as no obvious experimental fact contradicts the current theories (General relativity and the Standard model of particle physics). Instead of experimental problems, M/String theory addresses the theoretical inconstencies between those theories. This could be seen as far-fetched (not to me), but for sure it will be difficult to imagine more questions afterwards.
Now, one should be careful with the idea of theory of everything for two reasons.
M/string theory progresses, but has not reached yet the point of predicting new facts to allow testing it. It is more in the state of 'consistent set of ideas' than in the state of a complete theory. It might occur that to produce predictions, it has to lower its ambitions to "theory of almost everything".
Past history tells that it already occurred that the Physics community thought collectively that "all was known (but a couple of details)" and actually the "details" led to complete changes (e.g. the creation of quantum mechanics & relativity).
Time will tell!
"Why string theory?", you ask. I can think of three main reasons, which will of course appeal to each of us differently. The order does not indicate which I consider most or least important.
Quantum gravity
A full theory of quantum gravity - that is, a theory that both includes the concepts of general relativity and those of quantum field theory - has proven elusive so far. For some reasons why, see e.g. the questions A list of inconveniences between quantum mechanics and (general) relativity? and the more technical What is a good mathematical description of the Non-renormalizability of gravity?. It should be noted that all this "non-renormalizability" is a perturbative statement and it may well be that quantum gravity is non-perturbatively renormalizable. This hope guides the asymptotic safety programme.
Nevertheless, already perturbative non-renormalizability motivates the search for a theoretical framework in which gravity can be treated in a renormalizable matter, at best perturbatively. String theory provides such a treatment: The infinite divergences of general relativity do not appear in string theory due to a similarity between the high energy and the low energy physics - the UV divergences of quantum field theory just do not appear. See also Does the renormalization group apply to string theory?
Restricting the landscape of possible theories, "naturalness"
Contrary to what seems to be "well-known", string theory in fact restricts its possible models more powerfully than ordinary quantum field theory. The space of all viable quantum field theories is much larger than those that can be obtained as the low-energy QFT description of string theory, where the theories not coming from a string theory model are called the "swampland". See Vafa's The String Landscape and the Swampland [arXiv link].
Furthermore, there are many deep relations between many possible models of string theory, like the dualities which led Witten and others to conjecture a hidden underlying theory called M-theory. It is worth mentioning at this point that string theory itself is only defined in a perturbative manner, and no truly non-perturbative description is known. M-theory is supposed to provide such a description, and in particular show all the known string theory variants as arising from it in different limits. To many, this is a much more elegant description of physics than a quantum field theory, where, within rather loose limits, we seem to be able to just put in any fields we like. Nothing in quantum field theory singles out the structure of the Standard Model, but notably, gauge theories (loosely) like the Standard Model appear to be generated from string theoretic models with a certain "preference". It's hard to not get a gauge theory from string theory, and generating matter content is also possible without special pleading.
Mathematical importance
Regardless of what the status of string theory as a fundamental theory of physics is, it has proven both a rich source of motivation for mathematicians as well as providing other areas of physics with a toolbox leading to deep and new insights. Most prominent among those is probably the AdS/CFT correspondence, leading to applications of originally string theoretic methods in other fields such as condensed matter. Mirror symmetry plays a similar role for pure mathematics.
Furthermore, string theory's emphasis on geometry - most of the intricacies of the phenomenology involve looking at the exact properties of certain manifolds or more general "shapes" - means it is led to examine objects that have long been of independent interest to mathematicians working on differential or algebraic geometry and related field. This has already led to a large bidirectional flow of ideas, where again Witten is one of the most prominent figures switching rather freely between doing things of "pure" mathematical interest and investigating "physical" questions.
Best Answer
The idea which is being challenged, though certainly not disproved yet, is that there are new particles, other than the Higgs boson, that the LHC will be able to detect. It was very widely supposed that supersymmetric partners of some known particles would show up, because they could stabilize the mass of the Higgs boson.
The simplest framework for this is just to add supersymmetry to the standard model, and so most string models of the real world were built around this "minimal supersymmetric standard model" (MSSM). It's really the particle physicists who will decide whether the MSSM should lose its status as the leading idea for new physics. If they switch to some "new standard model", then the string theorists will switch too.
Whether they are aiming for the SM, the MSSM, or something else, the challenge for string theorists is, first, to find a shape for the extra dimensions which will make the strings behave roughly like the observed particles, and then second, use that model to predict something new. But as things stand, we still only have string models that qualitatively resemble reality.
Here is an example from a year ago - "Heterotic Line Bundle Standard Models". You'll see that the authors talk about constructing "standard models" within string theory. That means that the low-energy states in these string models resemble the particles of the standard model - with the same charges, symmetries, etc.
But that's still just the beginning. Then you have to check for finer details. In this paper they concern themselves with further properties like proton decay, the relative heaviness of the different particle generations, and neutrino masses. That already involves a lot of analysis. The ultimate test would be to calculate the exact masses and couplings predicted by a particular model, but that is still too hard for the current state of theory, and there's still work to do just in converging on a set of models which might be right.
So if supersymmetry doesn't show at the LHC, string theorists would change some of these intermediate criteria by which they judge the plausibility of a model, e.g. if particle physics opinion changed from expecting supersymmetry to show up at LHC energies, to expecting supersymmetry only to show up at the Planck scale. It would mean starting over on certain aspects of these model analyses, because now you have changed the details of your ultimate destination.