Probably pin 7 because I don't think a 74HC00N can sink enough current to damage an LED, at least, not quickly.
What value was the resistor (that you _connected with the LED_) and was it in series with the LED (or in parallel)?
If it's a standard LED for indication, then it's probably designed to run continuously at 20ma (though some are spec'ed as high as 30ma). So, it's important to limit the current to 20ma or below. This is tricky in the configuration you are using. So, best bet is to target a current of 15ma, to give it a little head room. (but, again, that's assuming it's a 20ma LED -- check the data sheet if unsure)
Different color LEDs have different forward voltages. And there is a range of forward voltages possible between even the same color LED and even between the same part number! This is what makes it tricky. You see, an LED is a "current device" and is best run with a current source, not a voltage source. The output of a 74HC' family logic gate is a voltage source. A crude way of converting a voltage source to a current source is to use a resistor in series -- the higher the value of the resistance the better this conversion is, with infinite resistance being the best (something that can be closely achieved with a constant current source or current regulator -- in which case it's a _dynamic resistance_). But, if you were paying attention, the problem with using a resistor for this, is the higher the resister value, the higher the voltage needs to be to achieve the required current (again, dynamic resistance is the solution to this).
BUT, because the current level is so low (assuming that you are using a common indicator type LED -- or even a low current super bright LED) you can get away with a resistor in series (and a poor conversion). Find out what the MINIMUM forward voltage is for the LED in question, and then use the following formula to determine the value of the series limiting resistor:
R = (Vcc - Vf)/I where Vcc is the voltage supplied to the logic gate, Vf is the minimum forward voltage of the LED, and I is the LED current when the LED is on.
You will need to get the LED current from the data sheet (for that LED). You want the max continuous current. If it's 20ma then use that if you are brave, or 75% of that if you're wise. If the spec sheet quotes 30ma, be suspicious -- decades back, I had several LEDs fail over time, that I ran at 30ma ('cuz the spec sheet told me I could).
As for testing an LED in a non "visual" way, when an LED is subjected to excessive current, the forward voltage typically rises. If the forward voltage tests higher than the maximum specified voltage (for the current that that voltage is specified at) then the LED is fried. But, a burned LED can still have a forward voltage within the specified range, so this is not a 100% test. Also, a bad LED can read as an open circuit -- that's pretty much 100%