Unfortunately, this is not available to the user right now. There are a lot of pain points in relation to this that are being brought up - including issues where a Triggered Send 'hangs' due to an error in an email, but no notification is set to recognize this error - so the queue keeps building up.
My best advice is to either work with support to see what offerings they can do with your situation (likely will require services, which will be $$$) or to try and set up your own monitoring system for checking queues and then troubleshoot the queues to see what caused the issue.
Here is a link to my simple Triggered Send queue limits monitoring script that may be useful.
From there, I would recommend looking at the queued records in the TSD queue link in Interactions in Email Studio and doing a 'send preview' of the Trigger on that record and see if there is an email error. You can also try the age old IT solution of 'turn it off and on again' via a pause/restart and see if that handles it too. Other than that, you really are at the mercy of support/services for solutioning.