I wrote about a similar one awhile back (DMC-T55) but didn't get a proper video. Now I found a DMC-S55 (late 2003) with the same issue and was able to record and clearly illustrate the issue in a video!
The Kenwood DMC-S55 started the new, round, speaker dock trend that was continued in the T-series and V-series.
Like the DMC-T55 I fixed earlier, there's an interesting hinge system that connects the battery door's positive contact to the board. Many other manufacturers (Sony, Sharp, etc.) just used a rotating, clamped on hinge to connect the battery door to the board.
Kenwood's design has a small "finger" that moves to a connected/closed position when the battery door is closed. What that means is you can clean the visible battery contacts (at the battery door and deeper inside), and still have a power issue.
I used the process of elimination (testing it on the charging dock) to determine the unit was still getting power, but something else was still amiss.
Opening up the unit, I could see the hinge mechanism had a bit of corrosion on it, blocking the electrical connection. One small swab with vinegar (and cleaning it w/alcohol) restored the connectivity and now it's working fine!
Of course, while I was in there, I gave it a bit of a clean and lube, but I'm very happy to have a working S-series (to add to my T- and V-series).
Please check out the video for the details!