Ah, you're talking the domestic side. You need a Priority Pass for that lounge, it's a pretty decent lounge and very quiet. I don't know if AV will give you an invite to it if you fly business class or not, they definitely do not for *G in economy. Airline operated domestic lounges in South America aren't that common. At least in the case of Peru the domestic flights are very cheap, in some other countries they are outrageously expensive on top of being treated like cattle.

As for hotels near the airport in LIM, there is one and only one. (at least safe hotels). There is a Wyndham on airport property in the parking lot near the bus depot. Aside from that one you'll need to go to Miraflores which is about an hour one way by bus, depending on traffic. The area near LIM airport is not safe, so you wouldn't want to go wandering around outside the airport property. There is also a Sheraton near downtown, but you'd have to go there by taxi, bus only goes to Miraflores. Then a few others scattered around like in Surco and Barranco, but those are farther than Miraflores.