Most species caught have 70 happiness (with the exception of certain species and N's Pokemon etc.).
>holding a Soothe Bell will apply a multiplier of 1.5 (rounding down) to only positive changes and that use of Luxury Balls will not reduce the effects of negative changes.
>A luxury ball doubles the happiness value of the Pokemon when caught. So normally, a Pokemon is caught with 70(depending on species) happiness but if you caught it with a Luxury ball, it would be caught at 140 happiness.
Walking 128 steps (50%)
>Several Pokémon evolve after being leveled up when their friendship level is at or above 220
Umbreon/ Espeon being one of them. So 220 is the target.
When Eevee is caught in a Luxury Ball, its base happiness will be 140.
It will take another 59 before it reaches 199 happiness, and the steps required for that are:
(128 steps (+1) x 59) / 1.5 (Soothe Bell) / 2 (Luxury Ball) = 2,518 steps (rounded up)
(128 steps (+1) x 21) / 1.5 (Soothe Bell) / 2 (Luxury Ball) = 896 steps
Total: 2,518 + 896 = 3,414 steps.
This is only for walking, and in Gen V.
NB: can someone please double check my maths.