It is almost always a bad idea to generate random variates one at a time in a time-sensitive construct in MM. This is a rudimentary way of doing a sim. run, with a million variates queued up:

    len = 15
    box = 2^len - 1
    cnt = 1
    Catch[Scan[(If[(box = BitXor[box, #]) == 0, Throw[cnt], cnt++]) &, 
       2^RandomInteger[len - 1, 1000000]]] 

Folding it instead of scanning may well be faster.

If you want to solve directly, say for the case of a box of six:

    d = DiscreteMarkovProcess[1, {
        {0, 1, 0, 0, 0, 0, 0},
        {1/6, 0, 5/6, 0, 0, 0, 0},
        {0, 2/6, 0, 4/6, 0, 0, 0},
        {0, 0, 3/6, 0, 3/6, 0, 0},
        {0, 0, 0, 4/6, 0, 2/6, 0},
        {0, 0, 0, 0, 5/6, 0, 1/6},
        {0, 0, 0, 0, 0, 0, 1}
       }];
    
    Mean[FirstPassageTimeDistribution[d, 7]]
    
    (* 416/5 *)

For the general solution:

    size[n_] := Module[{mp, sa},
       sa = SparseArray[{Band[{2, 1}] -> Range[1/n, (n - 1)/n, 1/n], 
          Band[{2, 3}] -> Range[(n - 1)/n, 1/n, -1/n], {1, 2} -> 
           1, {n + 1, n + 1} -> 1}, {n + 1, n + 1}];
       mp = DiscreteMarkovProcess[1, sa];
       Mean[FirstPassageTimeDistribution[mp, n + 1]]];
    
    size[20]
    
    (* 3234139734016/2909907 *)

Which takes only a fraction of a second even on the netbook I'm on right now.