It turns out that:
I don't know yet.
In attempting to solve it, my conclusion was flawed. I'll leave the first steps here to help anyone else along, because maybe there's something useful in there.
WARNING: I was having trouble spoiler-tagging the explanations with the images, so nothing below this point is spoiler-tagged, if that matters to fellow solvers!
Let's start with a not-to-scale sketch, filled with what we know:

Completing sets of 180° gives us a little more information:

Now we have a few more angles we can solve for. Let's call them G, H, I, and J to simplify the next step of math.

Based on what we know of the other triangles, we have four variables and 4 equations:
G + I = 170
G + H = 180
H + J = 140
I + J = 130
We can solve these as:
G = 170 - I
H = 10 + I
J = 130 - I
Which has infinitely many solutions that could make sense in this figure based strictly on a sketch. It may be possible to determine with some geometry beyond what I'm using, but I think that without being able to calculate the sides with proper "sin(x), cos(x)" calculations, it may not be possible.
My prior solution, which is no longer conclusive:
One solution is:
G = 140
H = 40
I = 30
J = 100

If this were the solution, we could be done. Looking at Triangle CDE, we see ∠ECD = ∠EDC. In an isosceles triangle, the sides opposite each of the equal angles are equal in length. Therefore we know sides CE = DE. And since CE is 5m, we know DE must also be 5m, QED.
The problem is that {G, H, I, J} = {145, 35, 25, 105} is also a valid solution set, and it completely ruins the isosceles proof while still "looking about right." Pinning it down to the previous ideal solution would take another geometric step that I don't know right now.
Feel free to supply it if you can!
As one other element that could help people, you can use the isosceles triangle property I mentioned above to find two pairs of equal lengths in the figure:

I wasn't able to turn that into anything useful, though...