Hi op, I had something similar last year and went to see a sort of life coach (she was actually my CBT therapist for PND but offered life coaching too). I needed to negotiate a payrise of about £20k at the time and was all over the place about it.
She told me that she saw loads of people in a life coach capacity and that without fail, men would just walk in and EXPECT that £20k but women tend towards the 'I know you're doing me a favour ' approach. She gave me the confidence to know I was worth it (it was the going market rate afterall, I wasn't asking for anything unreasonable) but the thing that made the difference for me was deciding, in a non emotional way, that if they didn't see fit to pay the role at the going rate, then I'd be happy to go off to a company that would.
In terms of the salary you should ask for. Consider the going rate for your skills and role, also consider how much it would cost them to replace you. Do they use agencies? What would the agency fee be? How long would it take to get someone new in and train them? Would they HAVE to replace you or would they likely just wedge your responsibility into other people's roles? Is there anything you do that they couldn't wedge into other people's roles? Consider the impact on THEM if you leave. This will give you a rounded picture of how willing they will be to negotiate.
Something else I did (which therapist said not to!) was suggest myself they do it in stages. I acknowledged that it was a difficult time for all businesses financially and suggested they give me half of the payrise on a certain date, and that I'd wait 6 months for the other half. I got that agreement in writing so I could happily go back after 6 months and remind them. In truth, they didn't then have to honour it, if wasn't legally binding, but that's a risk you take.
One other thing I did, and wouldn't recommend (!), is i totally lost my shit with HR on the lead up to the negotiation. They weren't involved in the payrise, I went to them in a pastoral capacity, seeking advice because my role was needing underpaid. They played the HR role, straddling the conversation diplomatically, and I went a bit over board with my reaction. Luckily, it didn't go against me, but it was a foolish move.
I think it sounds like you're in a strong position if your boss themselves is suggesting you talk about pay. Take that as a vote of confidence, get your facts straight about the market, consider wider factors about how inconvenient it would be for them to lose you, have a bottom line in mind, don't be afraid to nicely mention you understand if they can't give you X, as you hope they'll understand that you wouldn't want to stay in a role which is underpaid, and have a back up suggestion in mind as to how they might stagger the increase if you can sense they're on the ropes but not quite going to give you what you want . All very professional and with a smile. It's a negotiation- you are entitled to play your part in it!
Good luck!