It looks very peaceful!
If this really is (was) a shade garden, then you may have to give up on the Shasta daisies and move to shade loving plants. This is going to be a challenge (I have a lot of shade in my garden, so I understand that, though in my case I am not constrained to white in the shade; my Moon Garden has the sunniest spot in the garden), and you may have to rely more on foliage than on flowers.
What you can grow will depend on just how much shade that area gets. If it is a partial shade garden, then there are some things you can try:
For annuals, there are always white impatiens.
For perennials, you may have to rely more on silver or variegated foliage plants:
(unfortunately this has blue flowers, but a *little* bit of color in the Moon Garden actually brings some life to it)
While gold does not show up that well in moonlight (or dusk and dark), I have found that a
little bit of gold (or chartreuse) brings subtle but marvelous color to the Moon Garden. (
Helichrysum petiolare 'Limelight'
http://www.monrovia.com/plant-... is good for this; I think there may also be one in lemon, too.)
For partial shade shrubs, there are azaleas, camellias, and rhododendrons. You might also want to try a white-flowered
Loropetalum, but I have found their flowers to be insignificant and not showy in low light.
You might also look at astilbes, Calla lilies (especially if you can get them with spotted leaves), and English primose.
Depending on whether or not you are snow covered in the winter, you can also try things like hellebores and white-flowered cyclamens.
Apart from the impatiens, most of these look to be spring blooming things. (I think there may be one more white-flowered azaleas which also bloom in the fall.) Therefore the bulk of your "white" may come from silver foliage or variegated foliage plants. Unless you live in an area that is going to be snow-covered in the winter anyway, you may have to be careful choosing your plants to have a good balance between dormant and evergreen ones, otherwise your garden patch is going to look very bare in the winter.
Hopefully this will give you some ideas of things to try.