My DS is 6, and like my DH and myself, was an early reader. He can read far above his age level--but just b/c he can doesn't mean he wants to. He strongly prefers picture books, which is fine by me. There are still plenty of picture books at the library that just have way too much text for him--he sees huge paragraphs and he loses interest.
DS does a lot of reading, although we don't have any sort of reading routine. We also don't limit any media, but DS actually does a lot of reading while watching movies (we always have the English subtitles on b/c we're both sensitive to loud noises and a lot of movies seem to have poorly balanced sound, so we keep the volume low) and while playing computer games. DS is a natural scientist, so when I go to the library, I'll check out books about a variety of sciences as well as a ton of fiction books. He picks and chooses from the large stack I bring home what he wants to read. Sometimes he'll just flip through, looking at the pictures; other times he'll read the whole book; and sometimes, he'll take one look at the cover, decide he doesn't care for the art style, and refuse to even open it. I'm fine with all of that, so he's never pressured to read.
DS knows I'm happy to read out loud to him (which I used to do all the time until he was around 4 and wanted to read by himself). He almost always chooses to read silently to himself these days, but when he just doesn't want to put forth the effort, he'll ask me to read to him while he cuddles up next to me.
Until recently, though, I'd say his favorite thing to read was the Lego Catalog. Or at least, that's what he got most excited about reading. He's a Lego maniac. We also have a subscription to Highlight's High Five, but DS definitely enjoys the hidden picture type activities more than the stories.
His current obsession is Garfield. I had no idea there were over 40 collections of comic strips! The Garfield books are the one exception to his reading silently. For some reason, DS always reads the comic strips out loud. He'll spend about an hour reading through one book, and then revisit it throughout the next week or two before he's ready to move on to the next one.
So my best advice is to try to get reading material that your DDs are interested in, no matter the subject or the format. And lay off the pressure. Most of the posts I read on MDC from families trying to cut screen time down aren't interested in getting their kids to replace screen time with reading--they want their kids outside playing. As a fellow bookworm, though, I can understand why you'd like to foster a love of reading in your kids. But I'm a firm believer in not having everything you read be high quality literature. I know I enjoy reading "fluff," so I assume DS does too. In fact, DS has zero interest in reading something just to learn. But he's happy as a clam if he wants to read something, and by reading it he happens to learn interesting things.
Anyway, DH and I spend a lot of time reading (from news online to library books), and we've always got lots of books around (some we bought, but most are on loan from the library). DS sees us enjoying reading every single day, and we've always got a good variety of reading material on hand. We've provided a reading-rich environment...it's up to DS to choose to read.