First of all my husband has taught me that to untangle anything (this is sailor knowledge applied to ropes) don''t pull on the working yarns (in this case) but take the tangled knotted part and gently pull it out from the middle. Take hold of the center between both hands and pull your hands apart gently. Repeat several times. This will loosen things up so that you can untangle. Untangle as much as you can and then do the untangling action again. (It may be possible to find a video of how to untangle yarn or rope by doing a Google search.)
If this doesn't solve the problem enough for you to continue... If I add in a ball in the middle of a project I just leave a tail and leave a tail of the new yarn and start knitting with the new yarn. No knots, and I don't care where I change yarns. Sometimes I prefer the middle over the edges.
Doing this will leave a little hole. I think the nicest way to close the hole is to do a bit of duplicate stitch working in of the yarn on the back side so that it closes the hole. Knitty online knitting magazine has a tutorial about how to hide in ends with duplicate stitch. I'd recommend you practice joining in a new yarn a couple of times on a small swatch and see how you do before you do it on your project.