I think you'll find it difficult for seeds to germinate through grass. The ground will be dry and impoverished. Better to slowly but surely remove the tatty grass, add some good organic matter and then add seeds or plants/bulbs.

Yes, I was afraid that would be the answer. As the border is some 100feet long, or more(!) I think I'll just stick to the strimmer and flymo!
Many, many thanks for taking the time to advise me.

It would still be worth tackling one end or the middle of the border. Remove a few turfs every day you can and stack somewhere grass side down, in a couple of years you'll have one very good loam. Once you've got an area big enough to work on dig it over, add anything you can to improve the soil and plant - probably best t start off with plants and bulbs rather than seeds. A garden does require work but it is well worth the effort.