No problem getting your fly line and backing reconnected. The "go to" knot is the Nail Knot. A 5-6 turn Nail Knot will actually bite into the soft coating of your fly line and it gets tighter with more pressure. There are quite a few tools you can use that make tying it much easier. We really like the Tie Fast Knot Tyer. But you can always use a small diameter tube like a little coffee stir straw.
There are quite a few good diagrams and videos on the web that illustrate the exact steps for tying the knot. One piece of advice: After you tie the knot, give your backing several strong sharp tugs to make sure it doesn't pull off of your fly line. If it holds when you do this then it will hold when you hook into the big one!