He looked at her diary in Paris while she was sleeping and as he leafed through it he saw an entry that said 'Paris/Lunch with [somebody]/Hotel [name]' and the name was the same as the one on the forged letter fragment sent to him through the post. He put that together with the fact that she had said 'We must stop doing awful things to each other' and then clearly clammed up over the idea when he asked what awful thing had she done to him.
He then deduced that she had forged the letter and raced back to London to rifle through her desk before she could get there. Found the stash of headed/footed paper from that hotel. Drew the appropriate conclusion. Also found the headless photo and much more evidence he could use in a divorce.
They end up in court (as seen at start of ep 1, I assume there's a lot more of that to come in ep 3, which I haven't seen yet) because back in those days you had to go to court to get a divorce. One spouse brought a divorce suit, the other spouse either defended it, ie denied whatever was alleged as grounds for divorce, or (much more commonly) didn't defend it, but even then the judge had to consider the evidence presented and decided whether it was sufficient to grant a divorce. Sometimes they decided it wasn't. Either way it was reported in detail in the press. Far too expensive and nasty for most people, so the vast majority of unhappy marriages limped on until one spouse died, although they were often separated by then.