find DJ

Let $|\overline {AE}|=|\overline {GK}|=|\overline {HJ}|=|\overline {FD}|=x$ and $\angle {BHG}=\angle {HJC}=\angle {JKF}=\angle {EGK}=α$ .

Then $x=\dfrac{10(1-\sin α)}{1+\cos α}=\dfrac{10(1-\cos α)}{\sin α}$ .

Solving this we get $α=30\degree\implies x=\dfrac{10(1-\cos 30\degree)}{\sin 30\degree}=10(2-\sqrt 3)$ .

So $|\overline {DJ}|=x+10\sin α=10(2-\sqrt 3)+5=5(5-2\sqrt 3)\approx \boxed {7.67949}$ .

Let $EA = a$ and $\angle JKF = \theta$ .

Since rectangle $AEFD$ and rectangle $GHJK$ are congruent. So. $KJ = AD = 10$ and $GK = EA = a$ .

$\angle JKF = \theta \Rightarrow \angle GKE = 90\degree - \theta = \angle EGK = \theta \Rightarrow \angle BGH = 90\degree - \theta \Rightarrow \angle BHG = \theta$ .

Hence, $\triangle BGH \cong \triangle FJK$ by $ASA$ criteria.

$EK = a \text{ sin }\theta$ and $GE = a \text{ cos }\theta$ .

$JF = 10\text{ sin }\theta$ and $KF = 10\text{ cos }\theta$ .

So, $EF = EK + KF = a\text{ sin }\theta + 10\text{ cos }\theta$ .

But $EF = AD = 10$ .

$\Rightarrow a\text{ sin }\theta + 10\text{ cos }\theta = 10$

$\Rightarrow a\text{ sin }\theta = 10 - 10\text{ cos }\theta$

$\Rightarrow a\text{ sin }\theta = 10(1 - \text{ cos }\theta)\hspace{25pt} \cdots\text{Eq. 1}$

$BG = AB - AE - GE = 10 - a - a \text{ cos }\theta$

But $BD = JF = 10\text{ sin }\theta$

$\Rightarrow 10 - a - a \text{ cos }\theta = 10\text{ sin }\theta$

$\Rightarrow 10(1 - \text{ sin }\theta) = a(1+\text{ cos }\theta) \hspace{25pt} \cdots\text{Eq. 2}$

Eliminating $a$ from $\text{Eq. 1}$ and $\text{Eq. 2}$ , we get

$10(1 - \text{ sin }\theta)\text{ sin }\theta = 10(1+\text{ cos }\theta)(1 - \text{ cos }\theta)$

$\Rightarrow \text{ sin }\theta - \text{ sin}^2\theta = 1 - \text{ cos}^2\theta = \text{ sin}^2\theta$

$\Rightarrow \text{ sin }\theta = 2\text{ sin}^2\theta$

$\Rightarrow \text{ sin }\theta = 0\,$ or $\,\text{sin }\theta = \Large\frac{1}{2}$

$\Rightarrow \theta = 0\degree$ (Not possible) $\hspace{20pt}$ or $\hspace{20pt}$ $\theta = 30\degree$ .

Putting $\theta = 30\degree$ in $\text{Eq. 1}$ , we can get the value of $a$ .

$a = 20 - 10\sqrt{3}$

$DJ = JF + FD = 5 + 20 - 10\sqrt{3} = 25 - 10\sqrt{3} \approx 7.679$