Like EKWF said (#comment229709), a Logitech mm50 battery contains a single 18650 cell. Its battery cell is soldered to a protection circuitboard, and surrounded by insulator skin (heat shrink tube). Its battery protection board connects to mainboard through three wires: black (battery negative), red (battery positive), and white (NTC thermistor between black and white, if I recall correctly, but check yours to be sure). As simply one cell comprises its battery, it can be re-celled fairly easily. This battery is discussed in some detail, with pictures, on iLounge. (Topic: Logitech mm50 & Pure Fi anywhere batteries)
Logitech mm50 has an inlet for an external power supply. Its power plug is a coaxial barrel, outer diameter 3.5 mm, inner diameter 1.1 mm (approximate), insertion depth 8 mm. (You might scavenge a perfect replacement barrel plug from Nokia equipment.) Its original power supply is nominal 12 VDC, 1.5 A, centre positive.
If you are looking to make a replacement power supply (repurpose an AC adapter and splice a plug), then your replacement MUST have regulated output. (Non-regulated "old" style wall warts, big and blocky with a hot transformer, output a wide range around nominal rating. A piece rated "12V" might output nearly 20V while not loaded. Power capacitors in mm50 are rated 16V.) I suppose a one amp supply could suffice, but not much less than that: mm50 draws more than one half ampere when quiet and idle and iPod is not present.