That's a great (and challenging!) goal. Hope that you update us on progress.
Two data points here.
Every few years, I decide that I should be able to do some amount of running as a semi-athletic person who grew up playing soccer etc. Every time I ease into it, try to be super conservative, and wind up being sidelined for a month when an ankle, knee, hip, whatever swells up to twice its normal size. My wife made me do a full out 12 minute effort a few years back and I managed 1.5 miles. Had plenty more cardio-wise, but the legs weren't cooperating. But it feels good to run fast, which was such a natural part of being a kid and I don't think that later in life we should resign ourselves to shuffling around at 13 minute miles (not that there's anything wrong with that).
My wife is an ultra runner, trains a lot and is very structured/consistent. With a coach, she has been working on increasing speed. He had her do, I think, the same 12 minute flat out test that I did. Her initial was like a 7:20 pace and after a year of training with speed work mixed in, she got that down to, I think, a 6:33 pace.
Good luck!