Very little will grow there, as the hedge will be taking up most of the nutrients from the soil, so it may be worth reconsidering what to do with that part of the garden. Maybe think about raised beds that you can fill with decent soil, and then have a search for plants which are happy in an eastern aspect.

Personally, I'd talk to the neighbour about removing the hedge, installing a fence and reclaiming both gardens.
If they absolutely refuse to remove it and force you to live with their unnecessarily high hedge, I would put down a path that would allow me to safely trim my side of the hedge. Add pots that can be moved about to ring the changes

If the hedge is over 6ft in height you may be able to get your local council insist it is cut lower or removed. Otherwise Urbanite's suggestion is the best one.

In your opinion, of course.

If you do decide to grow something in that bed the soil will need to be improved with lots of compost etc. I'm trying to remember what I had in a similar situation - nothing spectacular but things did grow. One thing was a Portuguese laurel which was kept clipped. A Mahonia would do there. You might get away with a camellia if you really improve the soil and check that its acid enough as long as the bed isn't East facing. . Helleborus foetidus. Several ferns would succeed there - Google Ferns for dry shade. There are some perennial geraniums too - look at Long Acre Plants site, they specialise in shade plants. For a bit of year round colour try Euonymus fortunei, silver or gold variations are available. You'd probably get some flowers from crocosmias and daffodils.
If there is room to widen the border even in one place so that you get more light and moisure your options would increase a lot.

Every comment on this site is an individual's opinion Rosierose... What are you specifically stating is 'my opinion'?

Thank you to everyone for replying.
As I said, removing the hedge is not an option. Our neighbours would never do that, they don't spend money on their garden; we've just paid to have the hedge cut, as it was so high and out of control, even though it's their hedge. On the other side of their garden they have conifers which have never been cut, and are about 40' high, at least. They cast a long shadow over our garden from mid afternoon onwards. But, heigh ho, we shall be living next door to these people for the rest of our lives, so we bite our tongues.....
Particular thanks to Steragram for such a helpful reply. Your plant list will be a great help.We have considered extending the width of the border. It would mean removing a flagstone path which is in front of it, and would be quite a lot of work.

Yes but you'd only have to do it once and it would increase your options. We've lived next door to leylandiis too, and know what a pain they are. With a wider border and improved soil you could grow some nice hellebores too, and perhaps some Monardas?

You're right, and it's on the list of possible solutions.
But first we have to build a pergola over some decking. That's this year's project.

So completely different now that we have pictures to see what is going on.
The obvious thing to do is to move the path to widen the border. Then dig in loads of organic material to improve the soil. As you already have the flagstones the only cost involved is the organic material and your time - which In my opinion would be a better investment than a pergola and decking.
You could dig the new bed as wide a you like, curved, waved - whatever. I'd put some climbers in the back, clematis or rambling roses to hide the half dead hedge.
There are plenty of lists of plants on the Internet - if you widen the bed and improve the soil then you will have much more choice.
Remember to leave access to the hedge so that you can trim it in future.

Yes the pic makes a big difference - you don't have nearly the problem I was visualising as there doesn't appear to a shade issue and your planting options are very much wider.
Quite honestly I wouldn't do anything until you're able to move the path, except possibly buy an evergreen shrub to put in front of the dead bit of hedge so it can be getting on with growing in a pot while waiting. The present space is just too narrow for anything much to flourish.

You have loads of lovely space there to do something quite dramatic - and think about breaking up that long straight line.

I too would give it priority over decking which will go out of style, need regular upkeep and not last fer ever - as well as having a space underneath where rats often go.
But don't let me put you off!

Thanks for all the answers. I appreciate the thought that went into them. Plenty of ideas for me.