It has been a long time but working with concrete was one of the things that convinced me college would be a good idea.
Use either fiberglass or rebar not both.
Is it a corner where it washed out? If so it shouldn't see a lot of weight.
If it is bearing the weight of a vehicle I would dig 16" down in the problem area and add 12" of crushed limestone. Dampen it and tamp it down with a vibratory compactor. Bring it up to grade with sand and tamp that. I believe they use a vapor barrier now but am not certain.
Get some friends together for the pour, you have to work fast and raking concrete is really hard work. You don't need forms, your driveway and ratwall should provide that. Do you have all the tools like a floatboat and edger? Finish is not too particular, it should be a broomed finish so it is not slippery.
Lastly you can either trowel a seam down the middle or get a masonry blade for your circular saw to provide a relief crack.
As for the issue of water from the street, I had the same problem but it went into the neighboring lot and then into mine. After putting in 5K worth of drainage didn't cure it I built a berm/ flowerbed separating my lot from the one next door. I did this by forming two lines of small boulders about three feet apart and backfilling with dirt between them.