Falling edge detector using BJT
Your basic approach is flawed - all you are ending up with is a short pulse instead of a long one. As several folk on here have already commented you need a circuit that has a toggling action - in other words a divider.

Consider this approach to your problem.



The first part is a switch debouncer (R1,R2 and C1). This cleans up the signal from the switch but produces a slow rise and fall.

The next part of the circuit is a pulse shaper. This is nothing more than a transistor switch or inverter circuit and  produces good rising and falling edges.

The final part is a 'divide by two' circuit made from bjts connected as a bistable. C2 and C3 are there to produce a short edge pulse input from the collector of Q1. D1 and D2 are 'steering diodes' which allow the negative edge of the input pulse to alternatively switch Q2 or Q3 off.

There are numerous articles on the interweb which explain in detail how this divider works if you wish to do some research.