If it were left down to me, I suspect our garden would look similar as I am not that interested. I like a garden to look nice, but I just CBA!
DH likes a nice garden, so this is one of his jobs to do. However, there are some times, like at the beginning of Spring, that it just needs a bit of extra attention after having been left over the winter.
At that point he just says "we need to do the garden this weekend. If we both take a crack at it, it won't take long." and we do, and it doesn't!
What would your DP do/say if it was presented as a non-optional thing to do - "We need to get out in the garden this weekend and get it so we can use it"?
As PP have mentioned, have your DD out with you. Depending on her age, put her in a buggy or play pen, or get her to help pull up weeds or put things in the bin for you if she's old enough.
Clear all the furniture off to the patio and take a strimmer and mower to the main bulk of the lawn area and edging, put the furniture back and de-weed the patio.
It's not a 10 minute job, granted, but it looks like you should easily get the garden more presentable over the course of a weekend, even if you took it in turns with DD. If your DP really doesn't want to do the garden, then he gets to look after DD while you do it. He doesn't get to do nothing.
As for the rest of the house, just do a bit at a time, as PP have said. I know it's annoying having someone that isn't interested sharing the space, but he will either have to learn that there are certain things he has to do, even if it doesn't bother him, or he doesn't have to stay.