Square deal: a solution

On Futility Closet, Greg Ross proposed an interesting puzzle that i reproduce in the following picture, using GeoGebra:

The result of the quest (calculate yellow area, or \(A{EIBM}\) is 4 inch2. Why?
First of all we must proof the congruence between \(T_{AEIBM}\) and \(T_{EKCJ}\), who is a square's quarter, and so \(A_{EKCJ} = 4\).
First of all I proof the congruence between

1. \(\widehat{LEM} = \beta\) (definition)

2. \(\widehat{IJE} = \pi = \widehat{ELM}\)

3. \(\widehat{IEL} = \pi + \beta = \widehat{LEJ} + \widehat{IEJ} = \pi + \widehat{IEJ}\) (using 2 and 3)

4. So \(\widehat{IEJ} = \beta = \widehat{LEM}\)

5. \(\widehat{IJE} = \pi = \widehat{MLE}\)

6. \(EJ = EL\) (construction)
   Following (4), (5), (6) (according to Euclide) \(T_{EJI} = T_{ELM}\)

The conclusion is very simple:
\(S_{EKCJ} = S_{EJBL}\) for construction.
Now, some calculation:

$$A_{EKCI} = A_{EKCJ} - A_{EIJ} = A_{EJBL} - A_{ELM} = A_{EJBM}$$

So we can calculate the requested area:

$$A_{EJI} + A_{EJBM} = A_{EJI} + A_{EKCI} = 4$$