Your code initializes with k and fluxdiff being numbers but after one iteration k and fluxdiff are lists. Your while condition specifies fluxdiff >0.0001. But {number} > 0.001 will stop the iterations. For the second iteration, your code crashes because k is also becoming a list.

This should do the trick

out = First@Last@Reap@While[fluxdiff > 0.0001,
      oldflux = flux;
      oldk = k;
      oldsource = source;
      iteration += 1;

      flux = 1/oldk LinearSolve[a, oldsource];
      source = b flux;
      k = First[oldk Total[source]/Total[oldsource]];
      fluxdiff = 0;
      fluxdiff = First[RootMeanSquare[flux - oldflux]];
      Sow[{iteration, flux, source, k, fluxdiff}];
      ];
Grid@out

It's a typo, the issue exists without the typo.