It was recommended to me that I try to cut most of the old headphone jack out with snippers before trying to desolder.
Here's the inside of the headphone jack exposed. We found the cotton my mischievous child had implanted!
Getting the old jack out was a bit tricky; we ended up needing to use a solder wick (for example shown here: https://youtu.be/HZLQzBOOQbw?t=1817)
Here's the PCB after the headphone jack was removed, revealing four through-hole mounts.
I purchased Digikey.com part number 137-35RASMT6CHNTRXCT-ND and it has connectors in the right spot but they're flush with the bottom of the 'black box' chassis of the headphone jack as hopefully shown here:
Using the ESD-34A tweezers shown here: https://www.amazon.com/dp/B07D6JXD7J we gently bent the pins 90 degrees so they faced down and could be properly slotted through the holes in the PCB. Since the digikey part was so cheap I purchased two of them so I'd have a backup in case I broke the first one. If you come across this post and need one for your own toniebox let me know and I'll mail the extra one to you!
Here's a picture of the new part soldered back on the PCB:
There's a metal protrusion visible toward the front of the new headphone jack, which I guess was also a fifth pin (?) that wasn't needed? I left it intact, I hope that was an ok decision.
One random note, as I was reassembling the toniebox, I realize I'd forgotten which three-pin connector on the PCB was for the battery and which was for the speaker.
Fortunately the schematic I found here https://github.com/toniebox-reverse-engi... identifies that the connector labeled K8 on the PCB was for the battery and the one labeled K9 was for the speaker. I guess this should have been intuitive since the speaker connector is closest to the headphone jack and the battery connector is closest to the input power jack.
I screwed it all back together and it works!!!!! I can hear audio clearly out of both channels on my headphones and if I unplug them the speaker automatically plays the audio.
Thank you to all of you!