Electronic – Purpose of T flip-flop in this circuit (LED)

I duplicated your circuit in Logisim (as an opportunity to do something in Logisim). There's nothing wrong with your circuit. There is something about Logisim I don't understand.

First off, the red lines are not lines in a high state; they are errors. One would expect this sort of error if two outputs were tied together. I did a bunch of breaking the circuit and tying lines high or low, and eventually, all the errors were "flushed out" and reconnecting the circuit normally produced the toggling it was designed to do.

Specifically, break the upper leftmost wire, the one that connects Q' to D, then connect D to a high or low source ("pull resistor" works well here), and toggle it until it's all green. Then, reconnect the feedback, and it will all work. Note that high and low are represented by green and dark green (?).

Pressing "Reset Simulation" will bring all the errors back. My guess is, that somewhere in the logic of the program, it has an "undefined state". These undefined states propagate through the gates to the extent that they don't "sort themselves out" the way real electronics do. Undef AND 0 should result in 0, not Undef. Same goes for 1 OR Undef.

Just in case this has been addressed in a later version, I'll note this Logisim is 2.7.1

Update: I "fixed" the problem (within the scope of this simulator, anyway) by inserting a NOR gate in the feedback path. Then connect a pushbutton to the other input. I replaced the original button with a clock signal (found under "wiring"). Now, pressing the button clears the error. (Resetting the logic brings the error back).

Here is a schematic for a typical T flip-flop. Every time you press the button, the output switches.

All you need to do is to build the logic gates! You can use diode-transistor logic, resistor-transistor logic, whatever. When you're all done, connect the inverted output (Q#) to the data pin and connect your button to the clock.

You might consider debouncing the switch. This can be achieved using a resistor and a capacitor.