Let Kylie's speed be v. She travels at that constant speed for t1, and so before Mile begins riding she is a distance d1 = vt1 from him.
Mile rides his bike at a constant speed 4v, which is 4 times that of Kylie. Let's say that it takes a time T for Mile to catch Kylie. During that time, Kylie continues to walk. So, in time T she is a distance vt1 + vT from the starting point. During this same time Mile has ridden 4vT. Given that at this time T Mile has caught Kylie, this means that they are at equal distances from their starting point. So, you can equate their distances:
4vT = vt1 + vT
Subtract vT from each side.
3vT = vt1
We can cancel out v.
So, 3T = t1
T = t1/3 = (30 min)/3 = 10 min
So, Mile can catch Kylie in 10 min. It is interesting that their speeds don't matter, but only their speed ratio matters.