I am not unschooling, however...
My daughter (just turned six) was sloowwwly sounding out three letter words last May. As of August, she was reading and spelling rapidly and fluently at a 2nd grade level. Almost at a 3rd grade level now. I taught her basic phonics during the last school year (K).. just consonent sounds and short vowel sounds. Over the summer she was allowed to do as she pleased (think radical unschooling).
What happened? She started playing an online computer game for kids called Roblox, but any online environment that allows chat would have done the trick. Club Penguin, etc. She really, really wanted to chat with the other kids she was playing with, so she would ask us to read the chat log to her and type in responses. After a week or so of this, I told her it was taking up too much time and I couldn't sit and help her while she played for hours. So she figured it out, and I just helped her by calling out the spelling for an occasional word she didn't know how to spell.
I know you said your daughter doesn't care for the computer, but what about using it in an apparently non-educational context? What if she signed up for one of the online games designed for kids and just .. played the game? Both Roblox and Club Penguin have very strong filters to prevent inappropriate chat from taking place.
Alphabetical picture dictionary?
Something with words and speech at the same time -- TumbleBooks, karaoke, closed captioned videos?
Low/High books? (Intended for older kids who struggle with reading.) The idea here would be to get some of these books that have subject material intended for older kids but are written at, say, a 1st-2nd grade level and don't read them to her, let her figure out what they say herself.
If she loves books.. and you're reading them to her.. maybe you should cut down on how much you'll read to her. I realize this sounds kind of mean, but that's exactly the situation that prompted my daughter to learn to read on her own over the course of a few months. I just reduced my reading all of the chat / typing things in for her over a week or so until I was barely helping at all. I genuinely didn't have time to sit there and help her all day, so she had to either learn herself (with only occasional help) or just not chat with the other kids. She chose to figure it out.