Electronic – Why do digital controllers need A/D and D/A converters

Even a PLL isn't going to preserve the waveform (including duty cycle) of its reference input, unless you add a second circuit to it to specifically do that. For example, the PLL could drive a one-shot whose pulse width is controlled by a feedback loop that compares the output duty cycle with the input duty cycle.

The best way to do this in an all-digital manner is to have one circuit that measures the frequency and duty cycle of the input signal (if those are the only two parameters you care about), and have another circuit that synthesizes a new signal with the same duty cycle and the desired frequency based on those measured values. (I have done something like this myself when building FPGAs that need to do frame rate doubling/halving on video signals.)

The output signal will in most cases be an approximation of the signal you actually want, and you'll have to decide how good that approximation needs to be.

It looks like your latches woke up in the unknown 'X' state, and the complement of X is another X. For your simulation to work you need a way to initialize the latches in a valid logic state, either 0 or 1. Sorry, but I'm not sure how to do this in Logisim other than adding a separate reset input to your circuit. Perhaps you can specify an initial state in the properties for each gate.

Of course, in real life the latch will wake up in either a 0 or a 1 state and the toggle will work as expected. This is a common artifact of digital logic simulations...unknown values propagate pessimisticly...and you soon learn that it's best to have an explicit reset for the storage elements.