Simplest way to produce a lagging pulse (latch trigger) after receiving another pulse (clock trigger)?
You did not supply a schematic, and that image thing is incomplete .. but anyway.. the maximum rise rate for the clock input is about 100ns/volt with a 5V supply, and setup time is only about 24ns, so a simple circuit can be used- something like this: 







simulate this circuit &ndash; Schematic created using CircuitLab

If you don't properly debounce the pushbutton switches you will have issues.