Taking" non paying" parents to court

[v 1461]

Hello,
After some problems happening since January with a family I am still waiting for my money .
To keep it short I had to give a family 4 weeks notice in Jan ( I posted some messages about it on this website at the time and got good help).
The reason I gave notice is because the mum lost it one day and shouted at me during pick up ( if she had a knife in her hand she would have killed me ),
All of that because child came in with contagious illness ( didn't tell me ) and I said she should have told me.
Anyway I gave notice and they never came back and refused to pay the £450 for the notice period .
It has now been more than 3 months although dad promised about 5 times that he will bring the money but don't . Today I texted reminding about money and his reply was a" thumb up emoji".
We have now about 5 texts with promises and no show.
I thought the thumb up was enough and I decided to go to small claim court. I know it will cost me.

Also on my contract it says clearly I charge £10 per day for late payment which is 100 days)
I was thinking of first giving a message saying if they don't pay the £450 I take them to small claim and also charge the late fees.
Sorry for long post
I hate money problem and think 3 months wait is enough
Thanks for any advice