Totally agree with Collaborate, put your own fence up next to the boundary fence. It will be yours to stain/grow things against it, repair if it fails. And will hide from view the crappy fence next door.
Unless there is something in next door's deeds that states they have to maintain the boundary then there is sod all you can do. They are not legally required to repair the fence, in fact they do not even have to have a fence.
Save yourself the stress, cough up for a fence you want and like. Awful but it will keep your children safe, no more falling fence panels etc Plus imagine this 6 months down the line, new owner, no money to repair fence, new owners have kid, he/she keeps kicking the football into your garden and comes into your garden through the fallen fence panel gap to retrieve his/her ball. And repeat. (this did not happen to me but my friend)
If you want to look into this further, wander on over to GardenLaw forum and see how stressful arsehole neighbours can be about fences/boundaries/hedges etc
You do have my sympathies but sometimes you have to do things that other people should. Plus you get to choose the style of the fence (silver lining)