You should be able to run 1.56 then, maybe faster if your mile is strong.
Do that 4 x 400 workout I suggested once a week for 4 weeks.
Also, do you do any vVo2max type stuff - ie training at about 2km speed. You can do 10-12 x 1 min hard and 1 min easy, where the hard is 2km pace and the easy is 60% of that pace. Or else 6 x 2 mins hard 2 mins easy at same pace. This will help your back end. Do it once a week too, and on a trail in joggers ideally. Your 2km speed would be around 2.50-2.55 I'd guess, so the 1 min reps would be 240-350m and the 2 min reps double that.
4-6 weeks of each of those workouts once a week. They are the money sessions if you ask me. Add in a long run/cruise intervals, shorter easier runs, a shortish relatively intense tempo of 20 mins, and some real speed development (sprints, short hills, gym, resistance work) once a week, and you'll be flying. Remember Amos and Rudisha are the best over 800 largely because their 400 times are the fastest.
Let me work that out - each week:
1 x speed development, can run up to 100m but close to flat out, but must include shorter sprints flat out, sleds, short hills, gym etc
1 x 800 speed session (4 x 400)
1 x 2km speed session (12 x 1/1 or 6 x 2/2)
1 x 40 min progressive run/cruise intervals
1 x 15-20 min tempo
1-2 x easy run
If you want a day off, ditch the 2nd easy run. Do the speed development when completely fresh, after an easy run or a day off. Don't work the same energy systems hard 2 days in a row. You can swap days 3 and 5 around. The shortish tempo should be hard enough to give you benefit but not so hard that you can't back up and train well the next day. I think with it structured like this your recovery should be ok. I make this 25-30 miles, so not high mileage type training.
Day 1 - speed development incorporating gym afterwards
Day 2 - 2km speed session
Day 3 - easy run
Day 4 - 800 speed session, 2nd gym session after this if want a 2nd gym session per week
Day 5 - 15-20 min tempo
Day 6 - 40 min progressive run/cruise intervals
Day 7 - off