Why did the game keep going?

Me and my opponent both got down to 1 knight each. The game didn't draw automatically due to insufficient material, and i was up by 6 minutes, so i decided to play it out then. My opponent ran out of time, and draw by insufficient material. Why didn't the draw come automatically? We both only had a knight and king. What was the game waiting for? 

A mate is still theoretically possible in that situation.  The draw then isn't delcared until one of the players runs out of time.  On this site, a draw is declared on timeout if their opponent has one of the following: lone K, K + 1 Minor Piece, or K + any number of the same colored Bishops (i.e., promoted and/or original Bishops which all travel on the same colored squares).  Different organizations have different rules for when/if to declare the game drawn.

So if mate was possible why didn't I win on time?

Like I said, different organizations have different rules on that.  If this were a game played in a FIDE rated tournament, you probably would have.  If it were played in a USCF tournament, I'm not sure if it would be a win or a draw, but it might be ruled a draw there.  On this site, it is a draw, as my previous post explained.  The reason it is done that way here is purely for ease of programming.

(1) The reasoning is flawed - you should have won on time, if the rule that "the game goes on as long as mate is still *possible*" is followed, which I believe it should be.

(2) In a USCF game, you are allowed to declare "insufficient losing chances" in positions where mate is possible, but requires a ridiculous amount of cooperation. The general idea is that this must be a position which a 1000 player can easily draw against a GM, which the above position qualifies for - in fact, even someone who just learned how the pieces moved probably wouldn't fall for the mate. 

1.) If a mate is possible, there is no reason to end the game early.  Once time runs out, then you use the agreed ruleset to determine the result.

2.) This isn't USCF, this is chess.com and it uses its own rules to determine draws. 

In the USCF, the "insufficient losing chances" rule leaves part of the determination up to the TD.  Two different TDs looking at a game might give different results, particularly if it is in a complex situation, which is why TDs also have the option to simply change the time parameters - such as introducing a delay or increment in a game where one does not previously exist, with the idea that a player who truly does have "insufficient losing chances" should be able to hold a draw using just the time of the delay/increment.

Programming a computer is different.  They could have given the server many conditions under which the draw is given, but that couldn't cover every possibility.  They could have given the server access to tablebases to help, but tablebases are limited to games with only a few pieces left on the board.  They could have given the server access to a powerful computer engine, but at the time they programmed this, they didn't have the resources to analyze every game on this site that ends with a timeout, and such an analysis would not give an instant result, anyway.  Ultimately, they decided to go for simplicity.  The ruleset they use often gives the same result as USCF rules do, but with much simpler conditions that are not as prone to interpretation as USCF rules are.

Of course it's not the USCF, but I believe the rules should be that flagging is allowed as long as mate is possible, not some arbitrary result that basically hinges on subjective factors (because the mate is "impossible" because no one would be that stupid/unlucky).

And it's actually very simple to implement, so I don't believe programming is the problem: I can figure this out pretty easily ---

If either side has any pawns or major pieces, or just enough mating material in general by minors (AKA any two minor pieces will work), it's no problem, which means that there are very few combinations to consider (under the initial assumption of no more than one bishop of the same color per side):

*If both sides have at exactly one minor piece, excluding having bishops of same colors, then mate is possible.

*If one side has one minor piece, and the other doesn't, it's an instant draw (which is implemented).

*Else - there is possibility for mate. 

The only potential issue you might see is if someone decides to promote to bishops of same colors. But then: treat the bishops as simply one bishop, and the result will always be the same, given the extremely limited material available. 

So all they have to do is use this algorithm, which even I could implement, being a relative beginner in computer science. 

BTW: there is the possibility that every single pawn in the position is locked up, and hence no mate or progress is possible at all, no matter the move.

But let's ignore these situations , as they would still be declared wins on time by the current algorithm anyways. 

AKA: My proposed rule, which are definitely more consistent than the current ones:

-If there is mating material, then flagging is possible, regardless of possibilities of mate.

By the current rules, you had this subjectivity with possible mates, and it didn't cover the tons of (rare) positions where mating material exists but mate is impossible. This rule easily gets rid of the first element. 

I mentioned this in a previous post, but I'll reiterate it here since you're talking about situations pretty close to it.  Also, I checked the post where this was all explained and I was incorrect; I'll include the link later.

Chess.com considers "Insufficient Material" to be the following:

Lone King

King and exactly 1 Minor Piece

King and two Knights

When one player runs out of time, the server looks only at the opponent's material to determine whether it is sufficient or not.  Other than that, this is rather similar to the algorithm you suggest.  Could they have done it the way you suggest?  Certainly.  Did they consider it? Most likely.  Why did they do it this way? I can't say for certain, but they probably figured that when you consider the pieces for both sides, you introduce the possibility for inconsistencies where the specific position gives a result that is different from the general situation (i.e., there exists a forced mate when the pieces are on specific squares, but with the same pieces almost anywhere else on the board, there isn't one).  That does still happen, but it probably does so less often with the ruleset as it currently is.

This is at least somewhat explained in the following post and its comments (which is now nearly 6 years old): https://www.chess.com/blog/webmaster/live-chess-gets-some-love

Before this, the situation was different on timeout.  I don't recall for sure, but it might not have had any kind of insufficient material draw on timeout.  I recall there were some staff-made posts on the forums asking the community for suggestions on how to include it.  This blog post was the result of their deliberations.