Charging small capacity battery
I've used the MCP73831 which is a cheap linear charger from Microchip. They come in flavours appropriate to what features you want - in other words you get an -2AT if you want 4.20V, terminating charge at 20% of set current etc.

Choosing the one resistor correctly gives you the charge current. It's a reasonably narrow resistor range giving a current on a curve. From 10k to 60k or thereabouts gives you 100mA to under 20mA. Either side of that range, it's not really accurate. That range seems to be in the charge range you are looking for though.