Sometimes when you're stuck on things, it's helpful to look at the mathematics of what's being asserted. For example, nowhere in Newton's three laws does "energy is conserved" appear.
Energy conservation does appear, however, when you have a system that behaves like $m \ddot{x}=-\nabla U$, for some function $U$, where $x$ is a position vector as a function of time. In this case it's a mathematical theorem that $\frac{d}{dt}\left(\frac{1}{2} m \|\dot{x}\|^2+U\right)=0$.
Though it's easy to get carried away and start talking about nature and systems and why some forces can be represented as $\nabla U$, in every regular mechanics book* I've read, this is what things boil down to.
*regular mechanics as opposed to higher mechanics. In higher mechanics one states that the action $A[u]=\int L(u(t),u'(t),t)dt$ tends to be minimized. From that it's a mathematical theorem that if $L(u,\dot{u},t)=L(u,\dot{u},t+t_0)$ for all $t_0$, then energy is conserved. However then your question becomes, "why does nature tend to minimize the action" or equivalently, "why must we use a function like $L$?" To which one must appeal to experiment! There are no proofs of energy conservation just as there are no proofs of Newton's laws!
The total work done by all forces acting on an object throughout the motion interval of interest is what the work-energy principle involves. It never says "no forces are doing work." And it doesn't talk about changes in potential energy. The changes in potential energy are involved in the work done by the conservative force attached to the particular potential energy. The bookkeeping of work, (force and direction and distance), can become tedious in some situations, but it works.
Let's say a person lifts a stationary box from a floor, carries it somewhere in the same room and sets it on a table. Let's also ignore air resistance, etc. The initial KE of the box in the table/room/floor reference frame is zero. The final KE of the box is zero. The total work done by all forces acting on the box is zero.
What forces did work on the box? AHA! The normal and frictional forces of the person's hands initially did positive work (force is up, motion is up), then more positive work as the person exerted a sideways force with sideways motion, and finally some negative work as the person exerted upward force to gently place the box on the table rather than dropping it. Gravity initially did negative work (gravity down, box moves up), and zero work as the box moved sideways (ignore the slight bouncing of the box while it's being carried) then positive work while the box moved down. All this work on the box adds to zero.
Meanwhile, the heart and brain and muscles were all doing work internal to the person, so they had to go get a soda pop and sit down and rest.
Best Answer
The kinetic energy is indeed the work you have done (assuming no change in potential energy), however this is also neglecting the effect of friction. In real life, the kinetic energy of the box is going to be constant if its moving at a constant velocity even if you are exerting a force on the box and hence doing work on the box. This is because in this case, the work you are doing is being converted into heat and sound due to friction. If we neglect friction, your box would keep accelerating as you applied a force and hence the kinetic energy would keep increasing.