It sounds like you don't know the signal locations very well, so you need first to estimate them and then, given those estimates, triangulate your position.
If you want some accuracy and realism, consider adopting a likelihood model for the signal strengths, finding the maximum likelihood, and making a gridded map of the location probability computed from the maximum likelihood estimates. The global maximum on the grid identifies the best estimate of the location and the contours (relative to the maximum) give confidence sets for that location.
A general likelihood model is obtained by positing a formula for the signal attenuation and allowing for error. You won't get very far with a completely general formula (with an angle- and location-dependent attenuation function), so you'll have to simplify. For instance, you might consider a "universal" attenuation function, call it f, so that if the source strength at a WiFi location x equals a then the expected strength at another location y is given by
z(y; x) = a f(|y - x|).
For example, you might consider inverse-square attenuation for which f(t) = 1/t^2 provided the distance t is greater than some small threshold. As another simplification, you might take the strength reading z(y;x) at location y for the source at x to differ from the expected value by a normally-distributed error; assume all errors are independent; and assume they all have the same standard deviation (s). The contribution to the log likelihood of a strength reading z then becomes
L(y,x) = -[(z(y;x) - a f(|y-x|)^2 / s^2 + ln(s)]/2.
The log likelihood to be maximized is the double sum of L(y,x) over all locations y and all sources x. It is a function of the unknown locations, the unknown source intensities, and the unknown standard deviation of the errors. It's straightforward to find the optimal standard deviation and optimal source intensities (take partial derivatives, set those to zero, and solve), but for realistic attenuation functions f you have a non-linear problem for finding the locations. However, in your example it involves only 13 parameters so you should be able to dump it into, say, a multivariate Newton-Raphson optimizer and quickly get a good answer. (The statistics literature is full of methods to solve these kinds of equations.)
If you additionally assume the second device has proportionally greater sensitivity than the data-collection device, it will make little difference in the model I have proposed (because the signal strengths enter multiplicatively). In fact, if you let the errors scale with intensity (so they have standard deviation a *s* rather than s) the difference between devices should be inconsequential.
In order to keep this simple I have skipped over some statistical niceties, such as the fact that this is a multivariate prediction interval problem, not a confidence interval problem. If the amount of error is not great (i.e., s is small), the difference will not be of much consequence.
Not knowing your budget i'll try to give you options.
First thing is you're not going to find an SDK/API that you can do all of the things you mention. It's going to take a combination of technology and customized programming to make it happen.
1) GeoDatabase:
Start out with a GeoDatabase you can store all the collected data in.
You have Oracle Spatial, MS SQL spatial, among others but my favorite-which happens to be free and open source, PostGIS.
2) WEB Data Services:
You can serve out the data via geo-web services in standardized REST, GeoJSON and SOAP protocols. These will create the layers of data your application will be consuming. The Microsoft (big daddy) of them all would be ESRI ArcGIS Server. But I prefer GeoServer or Mapserver again, free and open source.
3) Visualization:
Your going to need an API to visualize your data. There's got to be endless possibilities. Google Maps, Bing Maps, ESRI Javascript/flex/flash and other apis, and my favorite OpenLayers. Your guest it, free and opensource. And these are only the web-based mapping API's you have a slew of other OS install-able options.
4) Authoring:
To help in editing, creating, viewing your data you'll need an application for that. The framework you choose might dictate this though. If you go with ESRI products you'll probably find yourself using Arc-XXX (arcmap, arcscene,...) products. But if you go the opensource free way you could use QGIS among others.
5) Customization:
Once you have all these tools you can use their API's to do what you need. You'll probably use at least a dozen other tools such as GDAL, JQuery, Python APIs, C# APIs to do what you want. But it's going to take baby steps.
Truth is: It can take you a while to get a hand of everything. I imagine learning the GIS as you go is going to slow down progress significantly. You're bound to make rookie mistakes. But if you don't have a timeline and this is more of a learning process go for it! If you have a budget thought, you might want to consider hiring someone to help you out. There's a lot to projections, datums, routing networks and just GIS in general. You'll need to get a grasp of the Geographic part in G.I.S. to do a good job.
Hope this helps your journey of building what sounds to be a great app.
Best Answer
Bing Map do this - http://www.bing.com/community/site_blogs/b/maps/archive/2010/12/07/bing-s-new-mall-maps-get-in-get-out-and-the-avoid-the-crowds.aspx
Using a mobile device will triangulate to locate the users position. (Just don't fall in any fountains whilst your looking at your phone!) http://en.wikipedia.org/wiki/Mobile_phone_tracking#Network-based
Example Mall Map: http://www.bing.com/maps/?ss=ypid.YN633x14747926&form=vemaps&vm=HCL-RooseveltFieldMall&i=1
note: not all mobile/cell devices have wifi.