Basic help/advice on under-cabinet LED lighting repair

[Defenestrar]

Very common connector. It clamps down onto exposed tabs on the LED strip at "cut" points. These would probably do the trick and not loosen over time. That was a quick search so there could be better options, or ones that will be more flexible if you get into more LED strip lighting (e.g. colored, adjustable white color temp, etc...)

 

[Defenestrar]

It's been too long since I learned LED lighting basics, but start searching for "LED strip lighting" as your primary key word followed by the modifier you want: tutorials/DIY/wiring/selection guide/controllers/power supply/etc...