The power connection on my JBL Onbeat Rumble stopped working also. I could usually jiggle it to get it going but this wasn't a reliable solution. I note there were a few people online that have complained about this but no solutions or disassembly guides. With nothing to lose I set about pulling the unit apart to fix it... here's hoping it will help others:
1. Under the orange base rubber there are five screws. You need to carefully peel off the rubber to get to the screws. Also under the sticker on the input/output section there are two screws to undo.
2. See photo for positioning of rear clips. You might be able to use an appropriate tool to undo them. I used brute force to open the case and the rear clips didn't break.
3. There are two more clips and two screws at the front of the unit.
I don't know how to get to the screws - possibly you need to unclip the front of the unit and grille. I just used additional brute force which undid the clips and snapped the screw mounts. The unit was then completely opened up into two halves.
4. You can now get to the input/output section. There are four screws to undo before you can lift it out. In my case I could see a cracked solder joint on the power input so I re-soldered it as best I could and confirmed the unit was now being powered up reliably.
5. To reassemble, reseat the input/output board, clip the unit back together, re-insert the 5 base screws + 2 input/output screws, then glue the rubber back on (it might be sticky enough not to need additional glue). The snapped screw mounts don't seem to matter as the remaining clips are strong. Just shake out any plastic pieces to avoid rattles.