I think this may work though I'm not sure why. I made it up through trial and error, aiding by [#18514](http://mathematica.stackexchange.com/questions/18514/finding-face-vertices-from-the-face-adjacency-graph). You'll need `nextCandidate` and `FindFace` from there.

First of all since you haven't provide some example data, let me generate some wireframe:

    mesh = MeshRegion[
      MeshCoordinates[#],
      MeshCells[#, 1]
    ]& @ VoronoiMesh[RandomReal[{0, 1}, {100, 2}]]

[![Wireframe][1]][1]

Then,

    graph1 = Graph[
      MeshCoordinates[mesh],
      MeshCells[mesh, 1] /. Line[{start_, end_}] -> {start, end},
      VertexCoordinates -> MeshCoordinates[mesh],
      GraphLayout -> "PlanarEmbedding"
    ];
    graph2 = Graph[
      MeshCells[mesh, 1] /. Line[{start_, end_}] -> start <-> end
    ];
    adj = AdjacencyMatrix[graph2];
    graph3 = AdjacencyGraph[adj, GraphLayout -> "PlanarEmbedding"];
    faces = FindFace[graph3];
    meshpolygons = MeshRegion[
      VertexList[graph1][[VertexList[graph2]]],
      Polygon /@ Most[SortBy[faces, Length]]
    ]

[![Filled wireframe][2]][2]

Then you can obtain the area of each polygon as follows:

    Area /@ (MeshCells[meshpolygons, 2] /. x_Integer :> MeshCoordinates[meshpolygons][[x]])

Let's see if the method works for your specific wireframe.

Some explanation:

- Basically your problem is about detecting smallest polygons from connected lines. This requires some clever algorithm. Thankfully we have [#18514](http://mathematica.stackexchange.com/questions/18514/finding-face-vertices-from-the-face-adjacency-graph). We can turn our wireframe into a graph and use the method there.

- `graph1` is the straightforward conversion. It doesn't have any crossings, but for some reason it doesn't work with `FindFace`. (I haven't taken time to study it.) Possibly the function requires its argument's structure to be of certain canonical form, so I convert `graph1` to an adjacency matrix first and then obtain `graph3`.

- `graph1` contains the actual positions of the vertices, so we need `VertexList[graph1]` in `meshpolygons`, but the **ordering** of the vertices used in `faces` is from `graph2`, hence `VertexList[graph1][[VertexList[graph2]]]`.

- `faces` also includes the largest, encompassing polygon. I assume that it has the most vertices and remove it with `Most` after sorting the list of faces by the number of their vertices.

  [1]: https://i.sstatic.net/j2DbJ.png
  [2]: https://i.sstatic.net/vaijA.png