It is taking forever...
There is no definite time span. You need to do as fast as possible. The easy way is to feed it, play some game in Amie, then feed it again
I just depends on how fast you can get it happy. This guide should help:
Source: ^__^ link above
Average time is roughly one hour, but some people claim to have done it in as little as 20 minutes.
It all depends on how much you play with it. Try this out:
Pet your eevee as much as possible until it stops showing hearts and gives out musical notes. After that feed it some poffins and then play with it. After playing with it pet it some more and give it more poffins. Do that a lot and you'll see that you will get max affection in no time.