Solving a HJB with a probability to transit to a new state
I would leave this as a comment but I cant. You are on the right track.

Once you know $V_2(k)$ then you can plug that into to the first hjb and solve.

To solve for $V_2$ you need to find the optimal $i$ as a function of $k$. Then plug $i(k)$ into the 2nd HJB. That will give you a second order ode. Solving that will give you $V_2(k)$ and you go to 1.