YANBU - tell her you cannot afford it, it is inconvenient and you said a weekend not a month. Why are you friends? You say she is great but list quite a lot of things which would put me off this friend.
MadamDS has it - you need to respond very quickly and firmly, lying if necessary. Something minor-ish that lends itself to the situation but which is otherwise incredibly painful, enough to justify no visitors - ingowing toenail, bunion surgery, gall bladder, something like that?
I would tell her that you invited her for a weekend so to let you know which weekend she is coming for during her months holiday. When she says she is planning on staying for a month I would say something like 'No really? We can't possibly put you up for a month because we can't afford it and I am having surgery during the month you are here'
You invited her for a WEEKEND not a MONTH - so stick to your guns!
Nobody invites themself for a month with their child when they were invited for a weekend. Either you're 'aving us on or she's a massive blagger and you need to tell her she cannot stay for a month. Unless you are happy for her to stay if she contributes financially - in which case ask her. You need to put your foot down on the internet. Just assert yourself.