Well, to begin with, your first number gang somewhat aglay. The land speed record that John Howard set on his bicycle in 1985 was 152.2 miles per hour, which works out to 68.04 m/s. So I can see where you got the 6 and the 8 from, but your little decimal point snuck over one place when you weren't looking.
I'll use your number to answer the question. If my solution turns out to be wrong, then it's because you copied the number wrong, and you'll have to work it out again with an initial speed of 68.1m/s.
Initial speed = 6.81 m/s
Final speed = 5.44 m/s
Amount of slowing down = 1.37 m/s
Rate at which the brakes slow you down = 3 m/s each second
Time needed to slow down 1.37 m/s = (1.37 m/s) / (3 m/s^2)
That's 0.457 second. (obviously absurd)
If initial speed = 68.1 m/s
Then amount of slowing down = 62.66 m/s
Time needed at -3 m/s^2 = (62.66/3)
That's 20.9 seconds. Much more reasonable.
By the way, John Howard's record was broken 10 yrs later, in 1995 .