Microsoft surface 2 tablet charging problem can be solved by using simple tricks. I have tried many times to solve charging issue.
1) shut down the tablet, unplug the power cable and leave it for few minutes, plug in back the power cable and let it charge for few minutes, turn it on.
2) shut down the tablet, press volume up + power button for 10 seconds, leave it charging for few minutes, turn it on.
3) Performed a full system recovery.