Unfortunately, the Surface Pro 2 (along with the other Surface devices) are among of the most un-repairable devices.
https://www.ifixit.com/Teardown/Micr...Teardown/18604
Attempting a power button repair, or any repair in fact runs a serious risk of breaking something critical.
All I can really think of right now is either going to a professional (I'm not confident you'll find someone who's willing to deal with it), paying a few hundred dollars (USD) to MS for a replacement/refurbished unit or purchase another device.
I strongly recommend going for a device that is easier to repair going forward. Dell, HP, Lenovo business-grade computers are usually designed to be dealt with in the field rather than MS and Apple's less repair-friendly hardware. :(