So you've kind of got a perfect storm of bad circumstances here in that you've already reached the point where SQL Server is unable to start. Normally at this point it's necessary to detach a database and move it to free space, but if you're unable to do that you're going to have to start breaking things and rebuilding.
If you have a mirror and a backup that is up to date, you're going to need to blast one unlucky database on the disk to get the instance back online. Once you have enough space, then take emergency measures to break any mirrors necessary to get the log files back to a manageable size and get them shrunk.
The above is very much emergency recovery and you've got to triple check that you have backups, transaction log backups, and logs anywhere you can so you don't lose any data.
Long term to manage the mirror you need to make sure that your mirrors are remaining synchronized, that full and transaction log backups are being taken, and potentially reconfiguring each database on the instance with a maximum file size where the sum of all log files does not exceed the available volume space.
Also, I would double check that your system databases are not on the same volume as your database data and log files. That should help with being able to start the instance when you have a full volume somewhere.
Bear in mind, if you are having to shrink your log files on a regular basis then there's already a problem that needs to be addressed.
Update: If everything is on the C: drive then consider reducing the size of the page file to get enough space to online the instance. Not sure what your setup is here.