Get some white spirit vinegar (either natural or chemically made acetic acid) and put that neat into a spray bottle with a good glug of washing up liquid and a tablespoon of salt.
Spray all your weeds with a good coating and within a couple of days (make sure it isn't going to rain for a few days) they'll go brown and wither and die. Then just pull/brush them up.
It's all biodegradable stuff so doesn't harm pets. The white vinegar smell will put them off going near it any way. White vinegar is also a very handy household cleaner for stains, grease and descaling too. You can even put it in the washing machine instead of softener. Works just as well.
I bought a 5 liter bottle of the stronger 40% acetic acid stuff. Killed my weeds dead and flat in 3 days, plus I descaled a bathroom tap spout which was losing flow. Still got loads left.