+118667 I drew it.
Obviously, that is not exact but I believe it is close enough to answer the question.
Of course you are not meant to do it this way. I was just having some fun.
Perhaps you can use my pic to help you with the exact geometrical calculations. 
Keep in mind that 180-135 = 45
+129849 Using coordinate geometry
Let A = (0,0)
B = (7,0)
C = (7 + 4cos 45 , 0 + 4sin 45) = ( 7 + 2sqrt 2 , 2sqrt 2)
D = (7 + 2sqrt 2 , 2 + 2sqrt 2)
E = (7 + 2sqrt 2 - (5/2)sqrt 2 , 2 + 2sqrt 2 + (5/2)sqrt 2) = ( 7 - (1/2)sqrt 2 , 2 +(9/2)sqrt 2 )
F = ( 7 - (1/2)sqrt 2 - 6 , 2 + (9/2) sqrt 2) = ( 1 - (1/2)sqrt 2 , 2 + (9/2)sqrt 2 )
G = ( 1 - (1/2)sqrt 2 - sqrt 2 , 2 + (9/2)sqrt 2 - sqrt 2) = (1 - (3/2)sqrt 2 ,2 + (7/2)sqrt 2 )
The line through HA has the equation y = -x
The line through GH has the equation x = 1 - (3/2)sqrt 2
So the coordinates of H = ( 1 - (3/2) sqrt 2 , (3/2) sqrt (2) - 1)
The distance from A to H = sqrt ( 11 - 6sqrt 2) = 3 - sqrt 2 (simplified using WolframAlpha)
Distance from G to H = 3 + 2sqrt 2
Total perimeter = 7 + 4 + 2 + 5 + 6 + 2 + 3 - sqrt 2 + 3 + 2sqrt 2 =
32 + sqrt 2
