It is unacceptable for you to have to live in this environment.
I would suggest that you write an email, with photos, directly to the chief executive of Jigsaw. Include a timeline of when you have called/contacted their call centre, what "repairs" have been carried out, and all the damp, concreted tiles and rotted skirting.
Ask them if they think that this is a suitable home for you and tour child, and if so, would they authorise photographs to be used on the home page of their website. Because if they wouldn't, then the website is misleading.
Insist on a senior person from Jigsaw, with decision making capability, come and visit your home to discuss what can be done to make it livable.
Include quotes from their website, and how your home absolutely does not match what they claim to stand for, despite your best efforts to be a great tenant and to work with their staff and operatives.
Sometimes highlighting nonsense like the situation you've been left in to the top person gets things resolved.
Copy your local Council's Chief Housing Officer and the Council Lead for housing (a councillor) in.
My sister and I had to do this over a botched kitchen refit, it took some serious persistence on our part, but we did get the kitchen floor lifted and re-laid, kitchen refitted, and the bathroom done by way of an apology. It was a proper mission, but ultimately we got our way.
HAs are great at doing nice shiny new builds, and lovely community initiatives, but they are absolutely accountable for maintaining ALL the housing stock to a decent standard.