sumDigitsSquared[n_Integer] := Total[IntegerDigits[n]^2]
stopValues = Join[{1}, NestList[sumDigitsSquared, 89, 7]];
iterate[n_Integer] :=
 NestWhile[sumDigitsSquared, n, Intersection[stopValues, {#}] == {} &]

numberOfDigits = 8;
maxSum = numberOfDigits 9^2;
loopVariables =
  ToExpression@Table["i" <> ToString[n], {n, numberOfDigits}];
iteratesToOne = Cases[Range@maxSum, _?(iterate[#] == 1 &)];
allIterators =
  Flatten[{Reverse@#, 9}] & /@ Partition[loopVariables, 2, 1];
maxCombinations = numberOfDigits!;

ssd =
  SparseArray[Table[n^2 -> numberOfDigits, {n, 9}], {maxSum}];

Do[
  variables = loopVariables[[;; digitCount]];
  iterators = allIterators[[;; digitCount - 1]];

  Do[ssd +=
    SparseArray[
     Total[variables^2] ->
      maxCombinations/
       Times @@ (Tally[PadRight[variables, numberOfDigits]][[All,
            2]]!), {maxSum}], {i, 9}, Evaluate[Sequence @@ iterators]],

  {digitCount, 2, numberOfDigits}];

onesCount =
 Total[Cases[
    ArrayRules[ssd] /.
     HoldPattern[{a_} -> b_] :> {a,
       b}, {_?(MemberQ[iteratesToOne, #] &), _}][[All, 2]]];

(10^numberOfDigits - 1) - onesCount