Mono and Astable multivibrators not working when connected together
Most likely when the LEDs turn on or off they cause some sort of transient on the supply voltage, what can make the monostable misbehave. Pretty significant transients can be created when long power supply cables are used due to the series inductance.

A negative transient on the supply may cause the voltage on pin 5 of the monostable to also drop. Once the voltage on pin 5 becomes lower than the voltage on pin 6, the monostable will reset itself and disable the astable circuit.

Things you can try to fix this:


Add a bulk capacitor across the positive and negative terminals of your power supply close to your circuit. This is a common practice in any kind of circuit to mitigate problems like this. Smaller decoupling capacitors next to each 55 IC can also help.
Add a capacitor between pin 5 of the monostable 555 and ground to filter the threshold voltage used in the internal comparator. Inside the 555 there are 3 ~5kΩ resistors in series. The external capacitor will form an RC filter, where the equivalent R is ~3kΩ (5kΩ//(5kΩ+5kΩ)).