Just a quick update after using my battery for a few weeks. I've yet to do an actual capacity test on the used 850mAh ipod battery I pulled for this project so I'll have to update the real measured capacity here when I do. However, as a rough measure of battery life, one full charge seems to get me through around three 40 minute albums listening at low to mid volume with minimum track skipping/fast forwarding. So roughly 120 minutes of play back per charge. Definitely not great, but this is actually right on par with Sony's claim that the original rechargeable would run for two hours. My guess is the 850mAh battery I pulled from my 2003 ipod video is quite diminished so I'll buy some new replacement batteries to make sure I am getting their full capacity.
Finally, I wanted to mention that I've added all the design files to the project page (or click here https://hackaday.io/project/165088-sony-discman-diy-lipo-battery/files). This includes the 3D printed case base and lid (I printed in PLA at 0.2 mm layer height, taking approx 40 minutes to complete) as well as the small battery contact pcb (courtesy of JLCPCB, 0.8 mm board thickness and ENIG gold plating on pads to ensure low resistance contacts and resistance to corroding, though HASL should work fine as well). I provide these files under the following conditions: I am not responsible to any damage from their use, feel free to modify them, and finally all I ask is attribution (link to my youtube channel and this project page) if you tweak them and repost somewhere online. Other than that go nuts!