There's no such thing as "the mempool", each node has its own mempool, and its entirely up to them how to manage it.
Of course, nodes can personalize their mempool to have no time limit, or to a shorter one than 2 weeks. But 2 weeks (336 hours) is the default time a transaction typically lasts in a mempool, this coincides with the default mempool size of 300mb, when this is exceeded transactions gets dropped from bottom to top, i.e lower priority transactions are removed and may need to be rebroadcasted. I use 'may' cause nodes do not have to use the default settings.This unconfirmed transaction might start reappearing at any point, as some nodes that still keep it in their mempool will broadcast it again.
I see no motivation for a node to rebroadcast an unconfirmed transaction on behalf of the sender. There are services which allows one to do this, but it usually does not help in hastening the confirmation of a transaction. 
