Yea 20 mpw is enough to run quite fast, but it needs high quality.
I think #1 goal should be to improve the bad endurance (only 28 min 5k atm). That can be done with cycling/swimming/walking etc. You can't just run 60+ mpw out of nowhere, but you need to stay physically active as much as you can. Your heart can't tell the difference between running and cycling and swimming, it will get stronger no matter what you do.
That process takes a long time, but improvements will be "persistent" and long-term. Then for running, with 4 runs a week, you could do that:
1x long run, building from 60 mins to 75/90 mins
1x tempo run, 20-30 min at a brisk pace
1x moderate run, around 10 min/mile
1x very easy recovery run, not longer than 45 min
After 1-2 months you could start alternating the tempo run with intervals, such as 4x800 or 10x400 focusing on relaxation. And keep doing bike rides, swimming, elliptical, whatever you can do build your aerobic fitness. These activities can be done with friends at low intensity as well.