I think that at 13 months old, there is little you need to be doing except keeping them physically safe, and letting them get on with exploring their own world. Respond if they come to you, kiss, cuddle and sing, read books, and if they are happily engaged in something, LEAVE THEM ALONE!!! (And rush off to have a pee or something else equally thrilling!)
I read a baby book by Penelope Leach, I think, and read a most amazing and freeing thing which she calls “benevolent neglect”. That is, you make sure your baby/toddler is safe and in a pleasant environment, then you step back and leave themto it. Oh yes!
Now I have an 8 year old and a 4 year old, and though the 8 year old is a little more hands-on and emotionally high strung, generally they are very independent people. I am essentially a single parent thanks to my husband’s job, and I would never, ever get a break if the kids didn’t let me take one. For the past two and a half years I have been able to have a lie in on a weekend, even without another adult in the house. True, I come downstairs to every single toy in the world, spilled drinks and crumbs everywhere, and two pairs of huge brown eyes assuring me that the orange juice on the carpet really was an accident, but its a small price to pay for two hours extra sleep!
The other thing I’d like you to consider as they get older is tidying up. Yes, you want your kids to do it by themselves eventually, and yes, if the place gets too bad it is overwhelming. But it is another symptom of micro-managing parents when they follow the rule of “one toy gets put away before another one comes out”. For a start it means you have a hundred little fights ALL DAY, and you have to supervise constantly. And unless it is a board game or something valuable, then I think it stultifies the kids imaginations. I came down this morning to every video in the house laid out as a road system (they have tracks, dunno what inspired the videos!), and lego buildings all around, and dinosaurs attacking the cars on the track, and my two boys stark naked, one with a paper crown and one with his blankie as a cloak, conducting the affairs of this universe. If I’d forced them to only have one thing out at a time, they couldn’t have done that. Right now the living room and back rooms are HORRIBLY messy but it is Sunday morning and we will have a blitz soon, so it doesn’t matter.
Good luck with it - enjoy them, love them, they don’t need any more from you than that.