Exhaust Fan in small half bathroom on first floor
You need to remove that grill and find out what's behind . Hopefully, there will be some sort of duct work that vents to the outside. You do not want to vent humid air into plain wall space. There are many fans to choose from so look around. I would tend to look more on efficiency than appealing. You can always spray paint it white. 

Here is a typical bathroom exhaust fan that has a 13.5" by 13.5" grill and 14.25" x 14.25" housing. It vents to the back.


here is a link to the site so you can get more info.

https://www.zoro.com/broan-fan-wall-10-38-in-506mg/i/G3064914/#specifications

you should also try to clean out as much of the debris from the duck as possible. It will help with the air flow.