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This article will help: https://www.gardeningknowhow.com/plant-problems/pests/insects/spittlebug-control.htm

If you can't use chemicals, you will need to dig them up or continually mow them down when they start to appear. Either method will get rid of them eventually without chemicals.

Personally, if I were you, I would post on a local message board that you have free daisies to anyone who will come dig them up. Normally, you will have more than enough people volunteering to remove them for you so they can plant them in their home.

Daisies don't need to be pruned like some other plants, but you can coax a second blooming through deadheading. This article will explain more:
https://www.gardeningknowhow.com/ornamental/flowers/fgen/deadheading-flowers.htm

This is not to say that you cannot prune them, however, but if you do so, this should be done in spring prior to new growth or fall, cutting the plant back to the ground.

If it is a large and old patch, it may be that they are in need of rejuvenation. The middle of the patch starts to die out if the plant is not regularly rejuvenated.

All you need to do is dig up the patch and divide the plant and replant part in the spot where you dug it up from. This forces the plant to rejuvenate and it will start to perform better. This article will help with dividing the plant:
https://www.gardeningknowhow.com/garden-how-to/propagation/propgen/dividing-plants.htm

Unfortunately, most blooming ''presentation'' plants are usually difficult and/or unlikely to be restored to their original beauty. They are intended to be enjoyed during the blooming period and discarded. This doesn't mean that you shouldn't plant them outside to see if they will bloom later, but usually, it isn't worth the effort. Most chrysanthemums, gerbera daises and similar beauties are breed, grown by chemical manipulation and ''packaged'' to fit certain criteria, and that doesn't include garden performance.