In a room like this you need a unifying color pulling everything togther. And, it needs to be light and natural, so it feels like sky when the other rooms look in here. You need to treat it like an atrium room.
Every wall / end walls to the white ceiling all need to be light and bright - here is a natural blue that will pick up on the blue in the rug just fine - http://www.benjaminmoore.com/en-us/paint-color/woodlawnblue
(but really, you should take the rug to another room - it is too busy and not "outdoor" like an atrium) the pavers are perfect.

My best advice is to paint ALL the woodwork / woodlook / ikea bookcases - take them outside, get a sprayer, roll on the primer first / kilz, and then spray them twice and let them cure. Use bm stratton blue - deeper furniture shade of same blue, blue-green. That takes this into the realm of new england cottage built onto over years . . you can use their leveling paint for wood cabinet additive if you want a great finish - use a semi gloss for that. The aluminum frames and every opening should be trimmed e to echo the one wall with white wood mullions. You can even do all the white trim and white doors in the stratton blue if I can help you see why you should . . it wil be well worth it . . If all the trim wood in here is the same stratton blue as the ikea and old wood chests and things lining the walls and all the walls are light blue . It takes a funky space and makes it historic instead of hippie.

You have a lot going on here visually because of what the room will be used for. You are going to have books and art, your table and chairs are great and eclectic - leave them as is but make everything else "appear" to be built in by using the same deeper tone - slant walls for some of the art (paint ALL the frames white like ceiling and distress the corners back to wood). When you are done, group the books by color.

Leave the ceiling white, but the end walls to the slant including the soffits and the two side walls need to be unified with the woodlawn blue .

Interesting. That beautiful blue is not too different from the tile on my kitchen counters. The house was built in 1947; er, that is, the first part of it was. LOL. Are you saying that you recommend the blue on the bookshelves, the old wood cabinet, and every wall, soffit and trim?

I would like to follow up with a few "after" pictures. We are so thrilled with our library. I could not (did not have time to) repaint all the bookshelves, so I modified libradesigneye's advice. I used Dunn Edwards' "Twizzle" and "Pistachio Ice Cream" for the two blues, which they said were their versions of the BM. Pistachio Ice Cream exactly matches my kitchen counter tiles.

I would also lighten your dark brown table and please center that rug in the room. Looks like a good start.

Thank you abb_ann. We live on a farm, and the library door (as well as every other door) is the "way in" for dusty farmers. The rug, a treasured family heirloom, requested to be off-centered to keep away from those feet!