Kazoo to Piccolo
From what I can see the Kazoo comes with a derailleur mount at the rear wheel which makes things easy if you want to go high end but to add gears, you need: a shifter, a cable and housing (long enough to go from the shifter, along the down tube, along the chainstay to the derailleur - just use a few "zip-ties" to hold it in place), a rear derailleur, a rear wheel with gears and a longer chain.
The least expensive solution is to find a 20" bicycle with gears at a garage sale. You get the shifter, the derailleur, rear wheel and maybe the chain for a fraction of what you would pay for the parts. If you want to go high end, you can put quality derailleurs on the Kazoo and the shifting will be much better as a result but the cost will be higher (again buying at garage sales will get the parts much cheaper but you need to know what you are looking for).
If you're mechanically inclined you can assemble it yourself or if not, drop by the LBS and have them assemble the parts. You probably have to drop by the LBS for the looonnng housing (even my spares bin didn't have that part) and a new cable.
Now whether you need gears or not is a separate issue:
My daughter and I ride out of a community named Hidden Valley, so every ride starts with a hill climb to warm up. So having her pedal in 46-26 means she provides almost double the assist that she could with the 16T rear sprocket that came with the bike...and I appreciate the assist. And the 46-12 means she can still be assisting when we are truckin' right along.
Now, it does take some (lots) of vigilance from me to ensure she is in the correct gear.
p.s. "thumb" shifters seems to work better than twist shifters...something I learned the hard ($$$) way...