The verb is かかる meaning to take time in this case.
In the sentence it is
So it means to not take time. The そう modifies it to mean "not so much" like the "that" in "It shouldn't take that long"
As in most languages, in casual speech things get contracted and abbreviated. So, かからない becomes かかんない。
So the whole thing is
By bike it shouldn't be taking/have taken that long
But with the nuance of someone taking an unexpectedly long time as opposed to a simple statement of how long they think the trip should take because of the だろうに.
Note that バイク exclusively refers to motorcycles and scooters in Japanese.