LAN has daily flights to Easter (IPC) from Santiago (SCL) and a few times a week from Lima (LIM). You can get to both LIM and SCL on Delta. They also have a weekly (?) flight from Papeete and I think you can get there on Air Tahiti Nui'i which is a Delta partner (maybe? not all sure about this last).

I've started toying with going to Easter Island and it seems (only an impression without rigor) that it may be much less expensive to do some local tour operator's tour rather than trying to put air and hotel together separately. I would look at that as an alternative for peace of mind.