OK. Sorry for the delay in responding; been busy. Apologies too if what follows contains errors - I haven't run the position through an engine.
Of the solutions proposed, Morgski's looks the best. In my initial & quick assessment of the position, I'd assumed 1 Nf5 failed! For example:
1 Nf5 Rd1+; 2 Ka2 Rcc1 and it's White who gets mated! But clearly I'd missed Morgski's idea: 3 Re8+ Kh7; 4 Rxg7 mate because I'd overlooked that the N controls h6.
So, 1 Nf5 Rg4 and, aside from GP34's amusing example on the theme, Morgski is correct to say that, with modest care, White will win in the end.
This provides the general point. Notwithstanding Black's tricks, White has such a weight of advantage that many moves bring home the win - except Rxg7 😀 So 1 Rd7 is fine; and 1 Ka2 is OK as well. White can jettison his N and retain a big advantage.
My personal choice, after I'd realised my good fortune, was to play the dull but effective 1 Re1 (because I'd missed Morgski's idea, as I've mentioned).
One final detail to round things off. In the diagram at the start of the thread, I've rolled the finishing position of the actual game back a move for aesthetic reasons. In fact, my opponent played 1...Rd1+; 2 Ka2, and I was about to award him a huge round of applause for the anticipated 2...Ra1+ - when he resigned 🙂 A pity!