I totally understand your reasons to try to make it work, especially if there's children involved... and I like how you are taking responsibility for the way your relationship has become.
Here's what I would do: Start fresh from the very beginning, start dating her again. Is it possible to get a babysitter to look after the kids? Take her out on a "spontaneous" date, plan something which she likes doing (did she like music/concerts, dancing, anything?).. Try to remember what made her happy in the past and why she fell in love with you. Be surprising! Be funny and be cool especially if there is negative feedback from her in the beginning... It might take her a while to adjust.
Start thinking of things to do immediately, but take your time and don't rush. There is definitely some PUA stuff you should apply to your relationship like leading or teasing her.
Anything that would make both of your lives more varied/exciting. Even stuff like going to the zoo with the kids. If you both don't have much time, it doesn't have to be something huge and time consuming, just arrange for someone to look after the kids one night and grab her, lead her out of the house and into a nice restaurant (check her timetable ahead of that

. You don't even have to tell her what you are up to.. Or just let her know to be ready at a certain time one night and in case she doesn't want to, don't be upset. Just be persistent! As you said, you have 6 months at least to turn things around and it might take a while until she warms up..
Sex is always good, just grab her at the right moment and "throw" her around a bit. Lead her.
But AGAIN don't be upset if her reactions are not as expected in the beginning. NEVER apologize for trying to bring more life into your relationship/family. If she doesn't jump straight away leave it and try again another day. I wish you all the best, take it easy and post updates!!