Two SQL Servers cross-mirroring each other?
What you are telling is possible. Since you are running Standard Edition of SQL Server you have to bear following things :


Standard edition supports mirroring ONLY in full safety mode.
You have to be careful about number of databases your are configuring mirroring for, since it is recommended not to have more than 10 databases mirrored to secondary server. Remember that - On primary server you would need 1 global thread and 2 threads per each of the mirrored databases. On Mirrored server - 1 global thread, 2 threads per mirrored databases and one additional thread for each mirrored database for every 4 processor cores. 2001270 KB article talks about the worker threads requirement.
You cannot use database snapshots on the secondary server - since you are on standard edition.



  I'm trying to purely have a failover system without significant additional licensing/resources.


I would suggest you to look into Log-shipping as an alternative solution for your scenario. It is easy to set up and is a good old proven DR technology. Minimum it allows is 1 min to take log backups. Logshipping does not have any hard limitations on the number of databases being logshipped.

Logshipping allows you to logship your databases to more than one destination which is a very good advantage over database mirroring. Also, logshipping has functionality to delay restoring of log files on the secondary server to protect against logical errors. Mirroring does not allow this.

Logshipping as a part of process provides you full backup and subsequent log backups (depending on the frequency you configured - minimum is 1 min) to allow you to have backups so that you can easily setup your environment form them. Mirroring does not.

At the end, a good Disaster Recovery strategy is totally dependent on - 


How much downtime  can your business survive? and 
How much data loss is acceptable ?