I want to keep my circuit on after a push button is pressed and released. Trying to do this I stumbled upon soft latch circuits, but they are too complex and I don't need the turning off feature. What is the simplest possible way to keep my circuit on after a momentary trigger, like that of a push button?
Electronic – Keep circuit on after momentary trigger
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Related Solutions
The circuit you posted won't work, unfortunately. There are a few problems with it, but one that is critical is the fact that you have the cathode of your LED connected to the GND (-) terminal of your battery. I'm not sure what are the resources have to build this circuit and what are you willing to purchase, so I'm posting a circuit that is quite simple and uses very basic parts. This circuit will light up your LED for a few seconds, but it won't have a steady luminosity and a sharp cutoff; instead, it will decay slowly. In case you need a steady light intensity with a sharp cutoff you can change the circuit a bit. But as I'm not sure about your needs, it doesn't make sense to go further with speculations. Try to play with this circuit and try to get a good understanding of what is going on. After that, you can start thinking about ways to improve it.
Your switch should connect the positive terminal of V2 to the positive terminal of V1, and V1 doesn't exist in your circuit; it was used to simulate the switch closing. When you close the switch, the voltage applied to C1 will create current through C1, R1 and Q1. Because the capacitor is in series with R1, it will slowly charge and reduce the current going through Q1's base which will also reduce the current through the LED. It's essentially a very simple high-pass filter. The values chosen have to be tweaked a bit depending on the LED you are using and the timing you want, but should be a good initial guess.
Have fun!
NAND gates with one input are useless, they only change the state of input signals.
Toggle flip flops are constructed like this picture from electronicshub.org 
This link is going to be usefull for designing a T flip flop.
Best Answer
You can use MOSFETs to build a soft latch circuit.
Q1 is a P-Channel MOSFET. Q2 is an N-Channel MOSFET, resistor values really depend on your requirement (you want a strong or weak pull-up/pull-down resistor?).
When you press the switch SW1, the Q1 will be driven low allowing current to flow, thus driving Q2 up and keeping the Q1 Gate low.
Please note that Q1 and Q2 both have Rds(on), giving a burden voltage. So for a voltage-sensitive load, this may not be suitable.