Have you been running any tools to view/track the temperature of the SSD to see if this is the problem? There are many available (AIDA64 (best, IMHO), HWMonitor, HWiNFO32/64, SpeedFan, etc.).
The replacement parts for the NUCs can be found here: Intel® NUC Spare Parts. Unfortunately, I do not see thermal pads included. I would suggest that, if you prove the SSD is overheating, you contact Intel Customer Support to see whether they can provide you with the thermal pad...
Hope this helps,