Blender doesn't allow a single face to have a hole like this. (Technically, a face must consist of a single sequence of distinct edges connected end to end in a loop. Your desired face has two such sequences, one for the inside and one for the outside.)
What you'll need to do is introduce some extra edges -- a minimum of two are required -- that connect the outer loop to the inner loop and then create multiple real faces to represent your single desired virtual face.
A fast way to do this (though it uses more interior edges than strictly necessary is):
- Delete all the faces from your object.
- Select all the remaining geometry (both the inner and outer edges),
- Select Edge -> Bridge Edge Loops
This will add multiple edges between all the inner and outer vertices and fill them all with faces.
A slightly more tedious method that results in the minimum number of interior edges (two) is:
- Delete all the faces from your object.
- Select a pair of vertices, one on the inside loop and the other on the outside loop, ideally in close proximity to each other. Use F to join them with an edge.
- Select another pair of vertices somewhere else (e.g., on the opposite side of the shape) and join them with an edge the same way.
- Use whatever method you can to select all the vertices for one of the two portions of the outer loop and the corresponding portion of the inner loop, selecting half your total shape.
- Use F to create a face, filling half your desired face.
- Repeat step 4 and 5 with the remaining vertices (plus the four shared vertices that are part of the added edges), to fill the rest of your desired face.