Don't use the wall faucet variety, UNLESS you want to purchase a new top for the vanity. Or return the vanity. Pretty simple. The round hole was meant to have a faucet. Not a "cover" which may also allow water down under, which may also ruin the vanity as there will be no containing water lines beneath the little hole. Bad idea,

You want very easy care so you won't have to be weeding, pruning, watering and feeding because it is hard work on a steep bank like that. Along the bottom edge, I'd plant a few soft, smaller plants that won't require trimming back and are OK to brush against. Many natives are great for tolerating poor soil and not needing to be nurtured or fussed over. Clumps of hardy grasses - enough of each to make an impact, say groups of 5-7 minimum. Perhaps red tussocks (Chionocloa rubra) for their bronze tints, and Silver tussock (Carex comans) for their graceful light green fronds, and Chionochloa conspicua, which is very pretty in flower, (like a miniature toetoe.) Mix in NZ iris like Libertia perigrinans for its bright orange spear like leaves and it will tolerate clay well. Marlborough rock daisy if it is a dry area. A little further back put in that lavender, smaller hebes for flower and scent, and possibly some of the many miniature flaxes which come in a range of colours. There are a few spreading prostrate coprosmas that are great in areas like this too. If you don't like the bronze rambler but it is growing well, think of it as the background to set off something that you like, so perhaps a spiky silvery plant like an astelia will contrast with it and be a feature, or a little bright green hebe, with white flowers would contrast nicely. Behind those, put the feijoas, full size flaxes, corokias, the larger hebes, pittosporums, and coprosmas.

I'm finding it a little hard to see the detail. Can you enlarge the top 2 plans?

A lovely rug on the floor would be nice. You seem to have some space, yet your table is pushed against the wall. You could consider a bigger table for centre of the room or making it into an office / desk / book shelf / comfy chair. You have lovely sun coming in. Even a bean bag and a small coffee table could make a difference. A medium sized plant would also look great. Set a budget and stick to it, re-purpose things if you can and / or buy second hand so you don't break the bank. I love the little air vent on the floor it looks like a little mouse house. Maybe you can get some removable stickers and make an art piece or at least a little mouse running to the hole.