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I did a similar installation and had no problem with the peel and stick. In most kitchens, the underside of the cabinets collects a film of grease and that's what keeps the LED strips from sticking. Clean the underside of the cabinets first and LEDs should stick properly. Also, I added PIV motion sensor connected to a small Arduino board. That allowed me to write a program to configure the motion sensor to keep the lights on for anywhere from 5-30 seconds when motion is detected. As long as there is a little bit of motion within that interval, the lights will stay on. When the motion stops, they turn off.
Since you are using a DC-to-DC module rather than a LiPo charging module, I suspect that you are damaging your battery every time you charge it. If the charging circuit does not cut back to a trickle when the battery is fully charged, the battery will be damaged.