The question is ambiguous.
1) If the question is how much time the chicken has to wait till it decides to cross the road, proceed as follows. The time of waiting can be represented as $T=\sum_{i=0}^N Y_i$ Where $Y_0, Y_1, \dots$ are the arrival times of cars (i.i.d $exp(\lambda)$, being a Poisson process with parameter $\lambda$), and $N$ is the number of cars until (not including) the first one to arrive after at least $k$ minutes. $N\sim\mathrm{Geom}(P(5. To calculate $E(T)$, use the formula of Total Expectation, conditioning on $N$. In the course of your calculations, you will have to use basic facts about infinite series of functions.
2) If the question is how much time it would take the chicken to cross the road once it started doing so, then again the data is ambiguous:
a. If it takes the chicken exactly k minutes to cross the road, then the expected time is k.
b. If it takes the chicken at least k minutes to cross the road, then we need to know the distribution of the length of time it takes the chicken to cross the road in order to answer the question what will come first: the chicken crossing the road or it getting hit by a car.
Note that in all cases, it is assumed the chicken is clairvoyant and can predict with certainty that the next car will arrive in no less than k minutes, which is quite an accomplishment, whether or not you're a chicken.