if you want to be able to go with 600m, you need to first learn how to kick, and from there learn how to sprint. a "kick" from 600m out isnt realyl a kick at all but more of a strong surge, at least at the highschool level in events like the mile and 2 mile. if you want to try the acceleratng every 100m, go for it, el g was a champ when it came to the 600m surge. thats how he finished his OG 1500m race in 2004.
the thing you need to know about accelerating every 100m is that its going to hurt like a bitch. you really have to cimmit to doing it. if done right, its not only going to burn off most competitors, but make it a last man standing race if you have anyone left with a 100m to go.
in order to be able to become "strong" enough to go wih 600m, you really just need to become "strong". long, hard intervals, anything that will build strength helps. tempos are always good. you need to work on your speed a bit though otherwise even with additional strength you wont have a great kick. 2:07 for 800m when you have already run 9:57 shows a lack of speed on your part, especially that 55 you cited.
strength equals speed isnt just a saying, its how it works. work on your strength to be able to use what speed you have at the end of a race. but in addition to that work on your speed as well. being able to go 28-29 on repeats shows you have some decent strength (im assuming this is morethan just 4 or 5 200m repeats)but you really do need more speed.
the way a kick works depends on two factors; anaerobic capacity (the amount of speed you have at your disposal) and aerobic capacity (the amount of strength you have at your disposal). you need to have access to anaerobic capacity at the end of the race in order to kick, but if you have had to dip into those stores before the kick has come, there isnt going to be as much of a kick. a kick is just energy saved for the end of the race. work on the strength but also the speed.