^^ strs8 has some good ideas.

What i'd also do is change your HIIT to 20sec work : 10 sec rest ratio. You'll feel it much sooner, but it will have a very beneficial effect to your recovery time within weeks. Last time I did a beep test it had improved from ~10 to over 13.4, that was from 2:1 tabata, not even doing running training.
Try to remember it's only 10minutes work. That's nothing - in fact you're resting for over 3minutes of that time! So just keep push push pushing, all the way through.
Your 1.5 mile run time is very impressive by the way. It shows you have the speed, just need to build up the recovery and oddly shows that you should have the higher-intensity endurance necessary already.
EDIT: instead of upping the number of sessions (you're not resting enough in between IMO if you do that), increase the overall session time. I.E. do your 10min HIIT, have a 10-20minute break, then do another set of 10min HIIT (which will feel fine for the first few minutes, but then you'll prrrrobably flatline. Just keeeeep pushing!) - this consequently will also do wonders for your lower intensity endurance. :p