The original battery (note my lovely swollen one here) has a special flex cable attached to it that's soldered onto the battery - if you're up for it, desoldering this cable is probably the cleanest/easiest way to install.

There are two contact points (second photo) in the middle that need to be de-soldered and re-soldered, and as you can see they are quite small and fiddly, and you don't want to bridge the two because they're connected to a live battery unfortunately.