What are my next steps for this bed?

Been there, done that.

 

I can't help with plants that like chalk @FelicityA , as we don't have that here, but others will be able to offer suitable suggestions, and you can also look at other gardens near you, to see what's thriving. If you don't know what they are, you can [sneakily!] take a photo and post on the forum and people will help ID them.
Having said that, many plants aren't fussy as to soil type or pH. You could look at shrubs like Choisya and Hebe for evergreens, to have some winter interest, and Cotinus and Sambucus for purples, although they aren't evergreen. Some Hebes have dark foliage too. You'd just need to check that they don't mind chalky soil. Philadelphus might suit too [if they're ok in chalk] and there are several types. It's a question of being able to maintain any of the larger shrubs though - many people hack them into 'blobs' which is never attractive. Other perennials will fit in to give height and colour, as well as seasonal, variations.
It's a very different climate from mine too, so many plants will survive far more easily.

I think it would be better to beef up the soil first, over autumn/winter, and then look at planting in spring. If you rush it, some plants may not thrive, which is always disappointing. Any soil type will benefit from the addition of organic matter, especially that rotted manure, but anything will help. You can just put a thick layer on top of what's there, and it'll work in over time. If the soil is really compacted, you could loosen it a bit with a fork first. You can add as much as you want really.
When you're ready to plant, those stepping stones will help you avoid compacting the soil further too, but you can also use old planks of wood or similar if you don't have those ready. As you mention your children going across the bed, it's also about safety for them, and I wouldn't be planting anything too airy fairy and flimsy - too easy for them to get trampled! Sturdier shrubs initially will be a better bet

I'm not keen on palms, but they're very popular 'seaside' plants, or for tropical looking gardens, and if you like it - then absolutely keep it. It's your garden after all, and if it's happy, it certainly gives you a bit of structure to work other plants around it. It might benefit from a bit of pruning/tidying though, to give you more scope for other plants.
I think it's also a good idea to spend the time over winter considering what you like or not. Then you can decide on positions for various different types of plant.