No, Li-Ion and NiMH batteries have different voltages and charging requirements. They are not interchangeable.

Thanks for the tutorial, especially the clear steps on disassembly. I'm sure I would not have succeeded without this guide.

By chance I happened to have a spare battery pack for my cordless phone that was an exact fit. The Uniden BT-1008 battery consists of two NiMH AAA cells wrapped together with a small wire lead. I cut the wire lead about 3/4 inch from the battery and solder-tacked the wires to the metal tabs that I peeled off the old battery, then wrapped the connections with electrical tape. There is just enough room in the channels under the battery to hold the wires and the taped connections. I'll post a photo if I can.

I highly recommend the BT-1008 for this repair because it is commonly available, cheap (about $3 to $5 for third-party equivalents), and has wires securely spot-welded to the battery terminals already. Using a pre-made pack is much safer than trying to wedge loose wires against individual cells and the plastic wrapping keeps everything secure and insulated.