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I'm sure there's an easier way to do this, but this method does work:
First, AB = CD = CG + GF + FD, so FG = 2.
By the Pythagorean theorem, in triangle AFD we get
and in triangle BCG,
Angles AFD and EFG form a vertical pair, so they are congruent and have measure
Similarly, angles BGC and FGE are congruent and have measure
Then the remaining angle in triangle EFG has measure
We can solve for the lengths of FE and GE exactly by applying the law of sines:
Let be the semiperimeter of triangle ABE, so that
Then according to Heron's formula, the area of triangle ABE is
