Hi Flips,
ok, first of all, You want to test it out:
1. Disassemble both satellites. There are 4 screws under the rubber lip on the bottom. The rubber itself seems to be taped with a 2sided tape. Be gentle though - You don't want to masacre the rubber lip if You want the sattelites to fit the bases again.
2. After taking the case off You see a battery (might look like a battery gone bad - the foil sleeve on my was puffed like a balloon; his seems to be generic li-ion/li-poly battery, so if You have problems with battery life it should be an easy replacement too), speaker itself and lot of electronic components.
You're looking for a part WL7010-D113 which is a wireless audio transceiver and looks like this:
Moduł Wi-fi 996580005346 - Dom jest w Twoich rękach - north.pl (3rd photo is the top view)
It is elevated a bit on some type of a hard foam riser IIRC (sorry, no photos because I don't want to disassemble it when it works without problem). The module itself has a pin socket, so removing it from PCB doesn't require any soldering. Be gentle when removing from the riser though - the glue is quite strong there.
3. Swap the element from the working satellite and turn it on (back button) - if that was the problem, the speaker should connect to the base without a problem (no flashing LEDs).
4. You have to find WLL7010-D113 itself. Some details:
5G Wireless Audio Transceiver / Receiver user manual details for FCC ID MDZ-WLL701X made by Amtran Technology Co Ltd. Document Includes User Manual Manual 20121213
fccid.io
I've found mine in a big spare parts online store in Poland for about 10$ a piece. I've also found a wholesale offer (around 0.1-0.2 $ per piece) on Alibaba.com at one time but it seems to not to be the case anymore. Now I'd probably ask around in local electronics shops or go straight to one of those local PHilips audio/Gibson warranty centers. Or just ask if the PL online store will send the items abroad.
5. replace a faulty module with a new one and turn on the set. It might not work the first time. This is because a set doesn't recognise a new part. You just need to do a factory reset and everything should be fine. Qute from the manual:
In OPTICAL mode, press and hold RESET
for five seconds.
» When the factory settings restore is
complete, the product automatically
switches off and restarts
6. Let it reboot and enjoy
Let me know if that helped!