Circuit to keep two LEDs on when buttons are released
The concept looks sound. 

The main thing I see is that your 5.6K resistor is way too high to guarantee a low level for a 'LS04. The 100 ohm resistors you used elsewhere are okay (but they'll get warm if the switch is held).

Suggest you use ~150 ohms for the pull down resistors and maybe 4.7K for the LED resistors. 

It would be better to use a spare inverter for each of the LEDs (so the output is low for LED 'on') and connect the LED + resistor between Vcc and the output. That's because you're loading the outputs with the LED and TTL can't source current that well. Then you can use 470 ohms or even less for the LEDs and they'll be much brighter.