Built a circuit on a piece of cardstock this weekend and it worked first try

I got into these a couple years back with copper tape instead of ink, and the thickness problem you hit is exactly why I switched. With tape you fold the corner over at the end of each trace so the sticky side meets the sticky side, and you get a solid connection every time. Took me three dead circuits before someone showed me that trick and I felt a bit silly for not thinking of it myself.

My version was much simpler than yours, just a couple of LEDs off a coin cell with a paper switch you press to close the gap. My daughter made one for a school project that lit up a drawing of a lighthouse, and she was prouder of that than anything else she brought home that year. One thing I learned the hard way with tape is that creases in the paper will crack the conductive layer, so I score any folds first and route the traces around them. Also found pressing the tape down firmly with something flat, like the back of a spoon, matters more than I expected. Since your kid wants a bigger speaker, she might enjoy swapping the touch pads for little drawings covered in foil, that way she can decorate each note herself.

I stuck with the ink approach longer than most people probably would have, and my problem was never gaps, it was drying time. I'd get impatient, poke an LED leg into a trace that was still tacky, and end up dragging a smear across half the drawing. Now I draw everything in the evening and come back to it in the morning. Slower, but way fewer restarts and no more ruined corners from smudging with the side of my hand while I work. I also started testing each trace with a multimeter before adding any parts. Takes a couple of minutes and saved me from the guessing game where the whole thing just sits there doing nothing. My lines wander a bit since I freehand them, so on longer runs I sometimes need to go back over a section twice, and checking first tells me exactly where before I've glued anything down. One thing I'd flag from experience is durability. Mine lived on the fridge under a magnet and the ink started flaking where the paper got bent at one corner. Still plays its little tune, but the next one is going on thin card instead so it survives being handled. My daughter keeps picking hers up and bending it, which is aparently what kids do with these.

Funny reading this, it took me right back to the old point-to-point wiring days when we'd build stuff on a scrap of wood with nails for terminals. Same idea really, just paper instead of pine. One trick from back then that might help your kid: a soft pencil, the 6B or 8B kind, laid down heavy makes a decent conductor for short runs. We used to patch a broken trace on a printed board with a pencil scribble more than once. Crude, but it works for little touch pads and she can draw whatever shape she likes. Give it a go before buying anything fancy.

Update: quick follow-up since some of you asked how round two went. We tried the soft pencil trick for my daughter's version, laid it on thick like suggested, and honestly it worked better than I expected for the pads. Still used the ink for the longer runs, but the pencil saved her alot of redrawing. Hers squeaks away on a bigger speaker now and she's already planning a third one.