If you could have this as an actual connection (without needing to worry about baggage etc), then you'd just have to go downstairs, go through security and go back upstairs to the departure gates (no need to go through immigration).

If this isn't possible, the process would be: go downstairs to Immigration, baggage claim, Aduanas (press the button to see if you get checked), then check-in, back upstairs to security, immigration (the departure tax is now included on the ticket).
11pm-midnight is the worst time for arrival at LIM, if you are unlucky you could be stuck in line at immigration for a long time. I wouldn't recommend taking the risk of missing your flight.