You can have a real day of pleasure and fun by hiring one of those mini-diggers.
Seriously though, if the weeds are that repetitive a problem, systemic weedkill the whole bank and repeat after 4 weeks, it will be a bit messy but best to start from scratch, after that get a good hardy grass (if thats what you want), like that canadian mix. Don't get impatient with systemic weedkillers (like I do!!!), they do a lot of unseen damage sometimes, if second application is too soon it kill too much of the plant before killing all the roots.
Alternatively, get loads of cottage garden mixed seeds, cover the bank and watch the amazing display of colours as everthing battles it out - only proviso to this is to make some sort of break between the bank and the rest of your garden.
If there are particular hardy weeds that cause problems (mare's tail, bindweed) then rub them a bit to cause some open wounds and put some systemic on them and a plastic bag (with systemic in it) round them to feed the systemic to the plant.
Ground elder though is just a persistant nuisance, you can try the bag trick but if there are a lot of different root systems (usually because it has been dug up in the past) then it is a constatnt battle. Sometimes a steam cleaner can sterilise ground elder out, but my experience is that the route systems are at two levels, top level is easy to sort out, bottom level always wins.
What I call Sticky weed (actually stickywilly
, catchweed, bedstraw, cleavers, goosegrass) is not a problem, just pull it up, it gets weak and gives up.