Our early 2011 MBP 13" stopped charging the battery and eventually refused to stay on even when connected with the MagSafe power adapter. We took it to an Authorized Apple Specialist, they determined that water damage was the culprit. The solution was to replace the logic board for $900. AppleCare would not cover it and having it repaired by Apple specialists would guarantee AppleCare could continue once repaired.
Ordering the new logic board and relevant parts came to a little over $700. The items arrived a couple days earlier than estimated.
From start to finish, the replacement process took about 90 minutes. Having a step-by-step guide with big pictures and color-coded notes was EXTREMELY helpful.
The guide listed the repair as being one of the most difficult. I believe its difficult because of the number of steps you have to take, not because the actual steps are challenging.
Be sure to read the relevant guides several times before starting. Make sure you have a clear workspace to lay out everything. Make sure you have everything you need on-hand. Only hiccup for me was not having rubbing alcohol to clean off the heat sink (repaired at someone else's house).
Finally, give yourself plenty of time and go slow. One moment of frustration and pushing/pressin