If you want real accuracy, note that the actual mile/km ratio is almost exactly half way between 1.6 and (1+sqrt(5))/2. So do both conversions and take the average.
(You won't get results as accurate as that suggests, of course, because doing the Zeckendorff + Fibonacci-shift thing only gives you an approximation to multiplying by (1+sqrt(5))/2.)
Given (official*) UK speed limits are always in multiples of 5, using the 8/5 rule would be quicker and work just as well.
* There are occasional places where private property has speed limits with weird factors, but they're so rare as not to worry about. (the one example I remember is when I used to cycle to work and took shortcuts through a sewage plant that had an '11 mph' speed limit, why? who knows.
How many km in 100 miles? 100/5 = 20, 20 * 8 = 160.
How many miles in 400km? 400/8 = 50, 50 * 5 = 250.
And 8/5 is 1.6 exactly, which is close to the "golden ratio".