LiIon Cell protection/charging circuit wtih AP9211, (Or other IC)
If you want to add a LED indicator, you can do it with a zener diode, or an opamp. For rechargeable Li-on battery (4.2 V max) you will need a low dropout voltage LED (red - 2V) and a zener diode with a very small zener voltage (Vz - 1.2V) and that's very tough to find.
So I will suggest an opamp variation. If that's your complete board, you'll have to add a voltage reference of aobut 4V for the comparator. The LED will lights when the battery is charged.
Anyway, you can try to find a charger IC with a LED output. If you want an easier handling, I would suggest not to use a QFN package. You can take a look at bq24091 - TSSOP package with LED charged output.