Does a mirroring configuration requires the database do stop receiving requests?
After you take your transaction log backup and restore it on the mirrored pair, you might have taken more transaction log backups.  If so, try to catch up on the backups then init the mirror.  At that point all the old transactions that are not replayed with be queued up and you will endure a DB_MIRRORING event on your log file which will stop the transaction log from truncating those transactions that have not replayed yet.  Do note if your network is really slow those transactions might continue to get queued up, it's critical to have an appropiate IO subsystem and network on the 2nd pair so it can keep up.

I'm not sure if you can in your environment, but if the transaction log backups happen too fast, perhaps then you have a IO subsystem problem you need to review on the 2nd pair or a networking problem.  Worst case, you could pause transaction log backups if it isn't going to cause a bad growth issue.  Otherwise you could look at using a DIFF backup to fully catch up then do 1 transaction log restore.  

Also, consider high performance mode over high safety mode if you can, but if you can't just make sure your network and diskIO are production quality.  Also set up monitoring for log delays, it will cause massive growth on the primary if you don't.