Lithium is very tricky to recharge. Rechargers on the market rely upon microcontrollers to quick charge, and then to top-off. Lithium batteries tend to be a fire hazard if charged too quickly or are overcharged. They sell little fire supressing bags to recharge them in to keep you from burning down your house due to a mishap. They can explode in a very exothermic fireball.
1) I'm suspicous of the 1K Ohm internal resistance value you've given. It sounds way too high.
2) One wouldn't normally charge a ~10 volt battery with 120 volts. To trickle charge you would use some supply, nominally greater than 10 volts, and include a limiting resistor, and cut-off voltage control for lithium.
3) Lithium ion cells output 3.6-3.7 volts. Where does the 10 volts come from?
4) Where do you get 120 volts DC? Rectification of the 120 VAC line gives you 168 volts.
5) Why don't you purchase a charger that won't burn down your mother's house?