I say NO to central Lima
I live in Lima half the year. My wife is Peruvian and we have a house there and travel back and forth all year round. Central Lima is not a good place for tourists to stay.
Miraflores is by far the tourist area and many speak English there. the Marriott is nice but very Pricey. There are many small older quaint hotels in the $50-75 per night range. Check on travelocity on there are many to choose from. If you need a specific question answered just PM me.
For sighting around the City take the Miraflora bus from Kennedy Park at night. Kennedy Park is in Central Miraflores. It gives you a tour of the city and downtown is very pretty with the lights at night. U can buy your tickets ahead of time. They have a sales booth in the Park.
Lima is a very nice city to visit and the Food is the best part of being in Peru. their food is becoming World famous and they have many Great dishes to try.