You should have called animal control each and every time the dogs got out. Since the dog is in your yard, call animal control and tell them that they need to come and get the dog. You need to start keeping a log of dates/times the dogs get out and call animal control each and every time. In addition, get an estimate for the cost to repair the fence and send it to your neighbor, with delivery confirmation, and demand payment within X days. If they do not pay within that time period you can sue them in small claims court (http://www.okbar.org/public/brochures/small-claims.htm). You might also want to call your homeowners insurance carrier and ask if you can file a claim for the damage through them and then they will contact your neighbor for payment.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.
To Attorney McFarland-Taylor's excellent advice, I add this: TAKE PHOTOS. Every hole, every incursion, every dug-up plant, every muddy pawprint on your clothes and your daughter's.
And report every violation to the police and animal control, with details about how "no-one is home" while the dogs run amok.
Your state and local bar associations can assist with referrals, if needed, and may have programs providing free advice.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
I agree with both of my fellow attorneys.
This is not to be construed as legal advice. I do not have an attorney client relationship with you.