I need a discrete and free way to cure lice infestation without my family finding out?

Buy cheap hair conditioner, lather it all over the hair, wait for them to suffocate then comb them out with lice comb. That's probably the cheapest method. Tea tree oil also seems to repel them and combine with lots of combing.
Otherwise you'll have to buy proper lice treatment in the pharmacy and apply according to package, but they can be expensive.