my small border is stressing me out...

Mine was when I started this season - I reckon all the rain last year, followed by the extended cold, and the dry spell was the cause. On the bright side, it breaks up quite well once you get digging at it with a decent spade.

 

It could be anything to be honest - lots and lots of bindweed around this area too.

Sadly, you can't make an omelette without breaking eggs, and to be honest I found the best way to go was to get set about the job, clear it all out (preserving bulbs etc) and then basically start from scratch.

I have a border on the left side of my garden that I have deliberately left alone this year so far, as I was concerned that there were loads of daffodils and tulips in there - - as it turned out, I had three tulips and five manky looking daffodils, so I am turning the whole lot over and making it into a bed for peas and beans - just think its best to accept that the plants that are in there aren't really worth a light, and get on with the job of making something out of the area.

 

Basically, yes - any you find when you dig the border over, make sure you put them back. However, if its only tulips and daffodils that were pretty poor anyway, you might be as well getting yourself some new ones. Daffodil bulbs will be on sale in the likes of Tesco in big sacks toward the back end of the year - cheap as chips.

 

It's a rented place so no doubt very uncared for by the usual tenants. Looks like the whole area needs glyphosphating and digging over from scratch.
As you're renting however I'd stick to getting some large planters or pots; plant those up and care for them and take them with you when you move.