It's funny. The common thinking is that for a speed 800 runner (a 400/800 guy like you), they're better off using their speed in the first lap and trying to hang on in the 2nd.
My kid's a 4/8 type, but did most of his fastest races holding back a little on the first lap and hammering it home, even negative splitting sometimes.
If you can go 56, then surely you must be able to handle a 64 first lap. Hold it round the bend, start winding it up a bit down the back straight, then turn it into a 200m sprint.
By the way I am not sure a 4/8 runner needs the heavy mileage most prescribe to run a good 800. I'd suggest that apart from your sprints/400/800 work, if you did these two sessions once a week each:
1. cruise intervals. This should be 5-6 x 1km at 8-10km speed, on short recovery. Do them on a trail in a park if you can. For you about 90 secs rest, so you might be able to do them n the 5 or the 5:30;
2. vVo2 max stuff, Bilat style. First week 12 x 60secs on (ie at your nominal 2km speed), 1 min off (jog at 50-60% of that pace). Next week 8 x 90 secs on 90 secs off. 3rd week 6 x 2 mins off 2 mins off. 4th week say 13-14 x 60/60, then 9 x 90/90, 6th week 7 x 2/2.
I would suggest these would be the only aerobic sessions you need. you could chuck an easy run in there one day, say 10 mins at 8 min per mile pace, 10 mins at 6:30 pace.
One session of 800 speed work, one session of 200/400 work, 1-2 of quality strength work. While I'm here - the 800 can be 2 x 6 x 200s at 32s on 30 secs rest the first week, 2 x 3 x 300 (48) on 1/6 min rest the 2nd week, 4 x 400 (64) on 3-4 min rests the 3rd week.
Seriously - do these sessions properly for 2 months and I can just about guarantee your times will drop. As I've said on here, I coached a 14yo sprint type 800 runner to 2.01 on roughly 10 miles a week (including w-up/c-down). It was all quality work though. And he was also a mid-52 400 runner which helps.