Preheat the oven to 325 degrees. Place a parchment round in the bottom of a 9 inch cake pan and spray with cooking spray. Set aside.
In a large bowl combine the flour, sugar, baking powder, and salt. Mix well.
Add the butter, milk, and eggs and beat with a mixer on medium speed and mix until well combined.
Add in the rum, coffee, and vanilla and beat on medium speed until well combined.
Pour the mixture into the prepared pan and bake for 50-60 minutes or until a toothpick inserted into the center comes out clean.
Meanwhile, heat the butter, brown sugar, salt, and coffee in a medium saucepan over medium high heat. Bring to a boil and allow the mixture to boil for 8-10 minutes or until it has thickened.
Reduce heat to medium low then add in the rum and vanilla and simmer for 2 minutes.
Poke holes in the still warm cake that is still in the pan. Spoon the syrup mixture over top of the cake and let rest 5 minutes or until most of the syrup is absorbed. Spoon more syrup over top and again wait 5 minutes. Repeat until all of the syrup is used.
All the rum cake to sit for at least 2 hours. Remove from the pan and serve with coffee.