I would first suggest just opening the device and see how badly damaged the sleep switch is.
Best case, the switch just needs to be reset.
Worst case, it might need to have a new one soldered on.
[http://image.prntscr.com/image/a1384e81f17745f1ade2129c0c113be6.png|Here's an image of where to find the sleep switch] if/when you open the device. Follow this guide to remove the logic board for inspection: [guide|35842]
You could possibly get away with desoldering the lid to the switch and inspecting the switch to see how damaged it is. Honestly though, I had a similar issue with a 2DS I fixed up. Reseating the outer switch to the inner fixed the problem.
Good luck!