The battery's kaput. The battery voltage falling dramatically once the charging current has been removed is the classic sign of a dud cell.
For a replacement, look for a 180 mAh lithium ion pouch cell (sometimes called a lithium polymer battery: they're the same thing). Does that circuit board have any components on it? In particular, is there a device with "DW01" marked on it? If so, that's a battery protection PCB. You'll need to find a battery with a similar protection PCB - they'll be called "protected batteries", logically enough.