Paying {4}{U}{U} and transforming Jacob Hauken, Inspector is part of the resolution of the activated ability. You draw a card and exile a card before choosing whether to pay.

You may play cards this way that were exiled with Jacob Hauken, Inspector before it transformed into Hauken's Insight.

Once Hauken's Insight leaves the battlefield, the player that controlled it may no longer play the exiled cards. If it's destroyed and somehow returns to the battlefield, it's a new object with no memory of the previously exiled cards.

Playing a card using the last ability of Hauken's Insight follows all the normal timing rules for that card. For example, if you play a land this way, you may do so only during your main phase while the stack is empty and only if you haven't yet played a land (unless another effect allows you to play additional lands).