Back when I was in my last year of secondary school some mates and I did LEJOG - 1 of the guys (well, boy I guess) only went as far as Ludlow as he figured it'd be too hard for him but he deeply regretted it - at the pace we set, using the route we took, it was pretty easy - our longest day was 70 miles, we had a few 2 or 3 day breaks built in, we stayed in youth hostels and avoided A roads whenever possible.
But, assuming you take a similar route to the one I did, you're gonna spend a fair amount of time walking if you're gonna do it on a fixie - some of those hills - steeeeeeeeeeeeeeeeeeeeeeeeeeeeeep! And the really steep ones have annoying crossroads at the bottom so you can't freewheel up the next one - nooooooooo - they make you stop at the bottom...look both ways...grrr
Still, fantastic fun and the feeling you get once you cycle into ...well, for us it was John O'Groats - is truly unforgettable.