Can't comment on YOUR work on this: can't see it!
Anyway, start with a diagram; like E = tangent point on AB, F = tangent point on AC.
You now have BD=BE, CD=CF and AE=AF.
Use pythagorean theorem and Law of Sines.
You seem unaware of the basics; like, no need to use pythagorean theorem to get BE and CF;
BE=3 and CF=4. And similarly, AE = AF.
Were you aware that the 2 tangent lines from a point outside circle to the tangent points are equal?
You would use Law of Sines to calculate angles ABC and ACB, leaving angle BAC = 180 - ABC - ACB.
Let M = inner circle center. Now work with the 3 inner triangles AMB, AMC and BMC.
Were you aware of this formula:
radius-of-inner-circle = 2[(area-of-triangle) / (perimeter of triangle)] ?
If we let x = AE = AF, then perimeter becomes 2x + 14 and area (using Heron's formula)
becomes sqrt[12(x^2 + 7x)].
This would be another way of solving, NOT needing the Law of Sines.
Hope this helps...