Depends on what the weeds are to some extent, but a glyphosate sprays like Round Up will be effective on quite a few.

Remove and dig out what you can and then any signs of new growth or anything you can not remove , spray with Round Up and see if that helps.

Thanks for the advice - but would round-up affect the plants a metre or so away? Anyone know the distance I need to leave between a plant I use round-up on and other plants?

Some weedkillers sit in the soil, some kill 'through the green', which means you spray the leaves and stems when they're actively growing - it'll only kill plants which have been contacted by the spray and is inert in the soil. Glyphosate is one that kills through the green, but choose a still day to spray it, or spray drift onto other plants will be a problem. Won't work on raspberries or blackberries or ivy, by the way, and anything else with woody stems.

I once killed some plants accidentally with weed killer drifting over from spray so what I do is pour some into a container and use a cheap paint brush to apply the weed killer to weeds that are near plants

Great stuff - thanks! What works for the woody stemmed variety?

For those you'd need a brushwood killer, such as SBK, but this must be used with great care because it sits in the soil and is toxic to all plants - usual method for use is to chop down, drill into or make cuts in the woody base, and apply SBK just sufficient to fill up the holes or cuts.