Sorry but I don't believe you can look after your kid all day and work from home at the same time. It's not a matter of privilege, it's a matter of practicality. If you have a full time job, you need childcare whether you work in your home or out of it. I don't think that's particularly controversial.
edit: If it lends any weight to my opinion, I guess I could also point out I work from home and that that despite your assumption I've been the only parent for two children <= 5 years old for the last 9 weeks.
