What is the standard way to choose a new leader in Multi-Paxos?
Using the terminology of the Leslie Lamport Paxos Made Simple a leader is implicitly chosen when phase 1 is executed and a node hears it has the highest proposal. The problem is not leader election as phase 1 is leader election; the trick is that any server can start and win a new election before the previous leader can make any progress. So rather than the challenge of getting a leader the challenge is not getting too many competing leaders duelling; yet not waiting too long before attempting to get a new leader.  See this answer that quotes where the original Paxos papers talk about having a stable leader after a proposal has been accepted.

The raft paper has a lot of detail about leader timeouts. Basically you should have the leader send heartbeats. These can be no-op values as described in the Paxos Made Simple paper. If a follower does not get a message for a while it can timeout and propose. You add a bit of randomness to the timeout to try to prevent simultaneous proposers. You should respond with nacks as mentioned in the Papxs paper stating the higher promise value. That way if you do get simultaneously proposers the nacks will suppress candidates as only one candidate will get the highest number to a majority leading to nacks to the other candidates inside of one full round of phase 1+2 messages to the winner. 

If servers stay up after the failure and the network stays stable a new leader should emerge promptly without too much delay or wasted messages on failed rounds due to duelling. The exact timeouts to set can be tuned to a given environment exactly as described in the raft paper.