Did I Make a Mistake on This Hand?

[Offsuit27]

OK, I'm playing in a $150 NL hold 'em tournament at Caesars. My stack is down to 3100 at a table of 10 players. The blinds are 400-800 and the ante is 75. A player in middle position limps. He has about 10,000 chips. The table has been solid, so I put him on JT+, 98s+ and 22+. It's folded to me in the big blind. I have K9o. I knew I certainly was not going to make him fold if I pushed, but for several orbits I had no opportunity to shove as my cards were truly awful, such as 85o. A push seemed marginal, but seemed like the best choice, so I went all-in. The limper called, turned over AJo and won the pot. I think I made the right choice, but wish I knew how to determine what my chance of success was against my opponent's range.

After that game ended, I felt my biggest mistake was in letting my stack get that low without taking any chances. So a few days later when I played in the same tournament again, when I had about 10,000 chips at a full table with the blinds 500-1000 with a 100 ante, I pushed all-in from early position with 99. A guy on the button called with AQo and I lost the coin flip. I was satisfied I made the right play, though.