Try completely unplugging the NUC's power supply from 'the wall' (A/C power) for 5 minutes. Before plugging it back in, flip the NUC upside down and remove the bottom cover (i.e. the one you removed for storage and memory installation). When you plug A/C power back in, you should see a green LED light up. In some NUCs, this LED is on the opposite side of the board, but placed so you see it glowing along one of the edges of the board.
If this LED (which is an indication that the NUC has standby power) is not lighting up, it means that (a) the power supply (a.k.a. power brick) has failed or (b) something in the board's voltage regulation circuitry has failed. All you can do in this case is try another power supply. If that works, great. If it doesn't, then you NUC will need to be replaced.
If you let us know what NUC you actually have (full model name), we can help point you to the right kind of replacement power to look for. In general, Amazon is a good source and you can often look for a replacement by searching based on NUC model name.
Hope this helps,
...S
