How to convert a battery pack to AC power?

Wife owns a very expensive piece of Japanese cosmetic device that uses a very special Li-ion battery pack but the battery died! No way I can find a replacement for the battery pack.  The battery pack available only has 2 wires (black and red) but not 3 wires (the yellow wire). I am afraid if I get a 2-wire battery pack, it won't work at all. There are lipo battery packs, but I don't think I can replace li-ion with lipo battery packs?!

I am wondering how to convert it to an AC power?

I have no experience doing this at all, so if you can provide as many details as possible, that will be great!

The battery pack:
3 wires connected to the connector: red, black and yellow; 7.4V, 750mAh

The AC adapter used for recharging the device:
INPUT: 100VAC 50/60 Hz 8VA
OUTPUT: 9VDC 500mA

Thanks a lot!

Picture of How to convert a battery pack to AC power?
adapter.jpg
The yellow may be the charging shutoff or a thermistor, it should be needed to charge. Doesn't mean you can't use the two wire. open the battery pack and post a photo.