Hello!

I have a 20 V analog signal that runs anywhere form 20-50 Hz and has a duty cycle that changes depending on how long a solenoid is needed to stay open. Basically is a DC pulse. I am attempting to essentially remove (split between two solenoids) every other pulse from the signal line, or in other words double the period, and split the duty cycle in half. Im looking for some suggestions on how to get this accomplished with hardware you could find in a school lab. Im currently using a voltage divider to run the clock off this signal on a J-K flip flop to adjust the signal (to no avail) then some Mosfets to deliver the 20 v signal power. If anyone has some ideas on how they would approach the issue let me know!

 

The concept looks reasonably OK to me, but put the solenoid between the MOSFET drain and the 20V pulse, not between the MOSFET source and ground.
Put a reverse diode across the solenoid to stop the back EMF doing any damage when it turns off.
Your divider on the clock input isn't quite right - 7.1k (where did you get a 7.1k resistor?) goes from the 20V supply to the clock input. 2.4k resistor and capacitor both go from the clock input to ground.
Is the signal noisy? Does it need filtering? If not, remove the capacitor because some clocked logic ICs like a nice sharp edge.
You haven't shown the negative supply to the JK Flip-flop - I presume you have one!

 

Need to know the part number for the JK flip-flop.
What are the values of the other resistors.

You have the clock voltage divider incorrectly connected as Ian0 noted.
As shown, you are applying the full 20V pulse to the clock input through the 2.4kΩ resistor.