If you can't tease the weeds out of individual perennials while these last remain in place then you have to pop the whole out with a fork and maybe put it in a tub of clean water, while you work the weed loose from the perennial.

And then put the perennial back in place.

Where it seems impossible to get a weed out of a perennial you may just have to give up and toss the combination, buy a replacement.

To prevent weeds from getting too big to be easily removed attend the border more often - there will be a certain minimum frequency of visits required in order to maintain the desired appearance and condition. Same as with the lawn for instance, with all the mowing and trimming involved with keeping that looking a particular way.

Can’t tell if you’ve edged the bed, this is essential for keeping weeds/turf out of flowers. You’ve probably got some kind of running grass, it’s just going to take diligence and persistence to get rid of this stuff/pulling every single perennial out, replanting, maintaining (as mentioned above). A lot of the time, weeds insert themselves (seed, runners) into planted perennials because the soil is better and looser and is just a better place to be.

The first picture is a clump of grass. Take a garden fork and loosen the ground around it. Then grab the grass and pull the clump out.

The last picture has a bedstraw plant (Galium) in the middle of another plant.. With your hand, go down the main stem of the weed to find the root and gently pull it out of the ground.

The second plant looks like Brassicaceae to me, not a Galium. But removal would be the same.