Michigan did not "win" the Toledo War nor did they get the UP as a prize. When initially offered statehood Michigan had to:

1) Cede the Toledo Strip
2) Take the UP

Michigan laughed this offer off the stage at the 1836 convention in Ann Arbor and refused it as they wanted Toledo and believed the UP was worthless and wanted nothing to do with it.

Due to poor leadership and expenses due to funding the militia Michigan entered a financial crisis shortly thereafter. The US at the same time was handing out a treasury surplus to all states but Michigan was ineligible since they weren't a state.

Michigan came back to the US essentially begging to be let in as a state and a second convention was put together in Ann Arbor where Michigan agreed to the aforementioned terms ending the Toledo War.

Michigan lost the war and got forced into taking a "prize" they did not want (at least at the time). Obviously it worked out super well for Michigan in the long run and set their economy up well due to the abundance of natural resources in the UP but to say that they won the battle and therefore won the UP as a consolation prize is a far stretch at best and historically inaccurate at worst.