Alice is moving towards a mirror with a velocity of $0.8c$. She sends a pulse of light toward the mirror when she is at a distance $L$ from the mirror and she times how long the pulse of light takes to come back to her. Bob is sitting at rest in relation to the mirror. What is the elapsed time from when Alice emits the pulse of light to when she receives it, as measured by Alice? Finally, what is the elapsed time as measured by Bob?
You don’t need special relativity's time dilation to calculate a time between events as seen by Alice, nor between events as seen by Bob. Alice and Bob are each in their own rest frames. They see what they see.
So calculate where the mirror will be when it and the light pulse converge to a point, then calculate where Alice will be when the light from there reaches her as she moves toward it.
Where special relativity comes in is questions like “what does Alice’s clock show to Bob when the light reaches Alice”.