If the soil stack is caste iron, ideally you should replace the whole thing but I can see you just want to do the feed into it at the bottom.
Get rid of the lead bit coming through the wall and replace it with plastic. There are a number of options for connecting to caste iron. The simplest one is a rubber collar which if tightened securely enough will be ok. There are other connectors though and it been a few years now since I was in the game so there might be something new out there.
You may need to grind the caste angled part of the soil stack to take the collar. It depends on what you use for the connection.
It’s a relatively simple job for someone with the tools, knowledge and experience who does this for a living. That doesn’t sound like you though so it may be best just to get someone in to do it.
Depending on what they find when they look at it, you are probably looking at a minimum of £300 or £400 provided there isn’t any reason to go down into the ground. Don’t forget, once the works done, there will be some finishing needed around where the pipe goes through the wall.