What sort of mulch did you use? If you applied something like crushed bark then you will get better results by moving it aside and digging over the ground below. You could add some new compost (all rotted down so it looks like dark dirt. You may be able to buy it from the garden shop or a landscape supplier.) You might also want to sprinkle some fertiliser. Lightly.

Sunflowers don't seem to like being transplanted so you're probably better to plant them into individual cells. Once they're showing above the ground give the little plants plenty of light, though not full sun, or they'll cook. If they're starting to look stretched out in one direction - they need more light. Remember to turn their container a little each day so they grow up straight. Don't despair if the early sowings flop. Try again, and see if the extra warmth and light that are around now give you a better, quicker result.

As your Cosmos did so well by just being sprinkled you could try that again, directly on the soil.

Beware of lily of the valley. Sometimes it can be just too successful! If your climate keeps it in check, great! Beautiful flowers; poisonous seeds, rampant roots. And it will love being fed/mulched.

Depending on the weeds you're trying to suppress you might also consider Hosta, Impatiens (as an annual), Dicentra, begonias for summer, bergenia, Liriope (if it's hardy in your area), Epimedium. There are heaps of lovely plants for growing in shade/dappled light for making a shady retreat over summer.