Why are Ka and Kb only used for weak acids and bases, respectively? The only answer which I found to this is that it's because strong acids and bases virtually completely dissociate, so the dissociation constants would be too large and hence aren't used. This doesn't really seem reasonable to me and I have a feeling there is another answer. Is there really another answer or is this the only answer?
This is absolutely not true. $\mathrm{p}K_\mathrm{a}$ values are frequently quoted amoung organic chemists for very strong bases and sometimes even for very strong acids. For reference, check out the Evans $\mathrm{p}K_\mathrm{a}$ table and the values quoted therein.