As I understand it, the procedure is drawing 4 balls out of 14. This yields 1,001 different combinations (order doesn't matter, so 1-2-3-4 is the same as 2-4-3-1). One combination is thrown out and the remaining 1,000 are assigned to teams. The last place team will receive 185 combinations (18.5%).
After the first drawing, they reinsert the four balls which were drawn out and then redraw four more. If the same team "wins" the 2nd pick, they just redraw until there's a unique winner. A few years back, I tried to calculate the exact odds but got a bit of a headache accounting for every permutation.
Assuming Ottawa finishes last and 'won' the top spot, then the readjusted odds would be (x/815) for the 2nd pick. So if the Kings had 13.5% (135 combinations), then their odds for the second pick would be 135/815 ~ 16.56%.
But for the exact odds, you have to calculate the probabilities of all the other teams potentially winning and the equation gets kinda ugly. It's easier just to sit back and watch the lottery with whatever lucky charm you have handy.