From never running to half Marathon trail run in 10 months!

So I'm back, after the Christmas and new years break and more eating than planned. I have realised I have completely fallen off the wagon. I am still trying to eat healthy and get my partner involved but find it hard to eat healthy the way I want, taking into consideration of his food preferences and apparent need for junk food. I have also found because he has been home from work for holidays I have not exercised once for fear of him seeing me. We also have a crawling teether who has decided to  sprout 4 teeth at once so moods are pretty gloomy. This is joined in by his terrible two's highly independent, strong minded and determined to push buttons sister. So at current being done physically for exercise is next to nothing. I am finding myself so mentally and emotionally drained just from the kids and house wife duties I tend to every day.

Because I have not been eating as healthy as I was, and my 2-3L of water a day has dropped I feel quite down and unmotivated within myself. So what did I do to fix this?

I decided to to do a half Marathon. But not just half Marathon. A trail run. So up and down hills, through different terrain. You know, not the sort of thing a couch potato normally goes, I will do this and train for it and get to the point where I will be able to run the whole thing ( I'm not worried about time it takes. Just to be able to finish and finish well, feeling accomplished)

So I have figured I am either an idiot who thinks I can do this in 10 months, going from nothing to 25km trail run on a mountain.

Another reason why I feel like this is a bad decision is our financial status. We are flat broke. Hubby works full time but having two kids under 2 and renting in a pricey suburb make it difficult for both finding budget healthy meals and being able to afford to get the right training equipment to do this both injury free and well. My current running shoes are hand me downs from my cousin who is male. So I'm a using old worn out shoes for the wrong gender, and all the websites I have read about starting to run say going to a professional shoe store to match shoes to the way I run are essential.

So overall, my goal is to go from this 

To being able to complete 25km (half Marathon) of this in 10 months

So what do you guys think? Is it possible for me, a mum of two young children, with absolutely no running experience and not as fit as I should be, to be able to train for a 25km trail run in 10 months?
If you want to help me reach my goal you can, I have set up a go fund me to make this actually possible but I need your help first.

https://www.gofundme.com/go-from-overweight-to-25km-trailrun