I have tape lights that have worked perfectly for 6 years. They were wired into a switch and I have no idea if, or how a transformer was used. They are not on a dimmer. Even you you turn off the switch, the lights will continue to light/blink for a tiny bit.
Dead power supply
LEDs pretty much don't fail. Power supplies are the achilles' heel of LED lighting, and the primary reason why LEDs fail at all. (#2 being the lead-free [bad idea] solder crystallizing over some number of decades). Unfortunately people cannot resist making [read: buying] cheap power supplies.
You'll need to find this power supply unit, identify its voltage and current output, and get another one with SAME voltage and SAME OR GREATER current output.
If you want a hand with that, get a phone camera with a flash, and shoot a pic of the label on the thing. Then you can edit your question to add the photo.
If your LED's are more than 2 years old, blinking usually means that a circuit in the driver portion of the led lighting unit has failed. Things like this one usually cannot fix.
Best to buy new.