Need help with logic gates

So I recently purchased some logic gates. I have the SN74LS08N AND Gates and I configured it as displayed in the image attached to this post. When I press both of the buttons down the LED flashes really quick and wont stay on. I'm not sure if this is normal, or possibly if it might have something to do with the type of chip I purchased. I'm looking for a way to make the LED stay turned on as long as both of the inputs of the gate are high. Suggestions
please!

Thanks in advance

Increase the pull down resistors to 4.7K or 10K.
Increase the LED resistor to 330 Ohms.
Look up 'switch bounce'.

Assuming your circuit is wired up as in the diagram (you may want to change the symbol for +5Volts).

You are trying to source too much current from the output of the gate.

High output source current is only 0.8mA.
You are asking for (5V - 2.2V)/220 = 12.7mA, assuming 2.2V Vf for your LED (Red LEDs around that).
As LarryD says, you are asking for too much.

If instead you flip the LED over, connect pin 3 to the cathode, and connect the resistor to +5 instead of Gnd,
the 74LS08 can sink 16mA, and the LED will be on unless both switches are closed.

A B AND Out
0 0 0 Low out = LED on
0 1 0
1 0 0
1 1 1 High out = LED off

The only way you'll get that to stay stable would be to use an AND gate (4000x series) at around the upper limit of 18v, then it would have enough current to drive your LED.

Change the resistor to the LED for a 1k one, it'll be dim, but it will stay on, thus confirmation.

or drive a transistor with the output.

CrossRoads:
http://www.ti.com/lit/ds/symlink/sn74ls08.pdf
High output source current is only 0.8mA.

I see it as 8mA.

You are asking for (5V - 2.2V)/220 = 12.7mA, assuming 2.2V Vf for your LED (Red LEDs around that).
As LarryD says, you are asking for too much.

Increasing the resistor to 330 Ohms, as I suggested above, will reduce the current drawn to around 8mA.
I'm successfully driving LEDs, with a 330 Ohm resistor to ground, from other 74LS ICs (and there's still sufficient current to pull an Arduino Input pin high). All 74LS series ICs (should?) have the same output characteristics, for compatability.

Where anyplace on that datasheet do you see IOH of 8mA?

So the problem is that the current that flows from the output of the logic gate is too small?

Thanks for all the help guys, I explored some of the ideas suggested and got much better results.

Yes, LS08 only source (drives high) with 0.8mA of current.
However, it sink (pull low) 16mA of current.

If you had gone with a74HC08, it gives you +/- 25ma. That's enough to drive most LEDs quite brightly.

25mA as an absolute max. Not a good idea to design to that.
Voh & Vol are likely to be well away from 5V or Gnd at that high a current also. Exceeding 4-5mA will cause the voltage to start moving away.

I've run them at 20ma driving LEDs for a long time. If he needs more current than that he can parallel two gates for a max of 50ma. Besides, most small LEDs of today are bright at 10ma.