I am unsure how to go about getting proof that Resharper will benefit the business.
If they asked for a business case, they're not asking for proof, just some kind of fact-based estimate of the likely return on their investment.
So, for example:
A license costs (say) $250 per developer, a developer costs (say) $50,000 per year.
A developer with Resharper costs 0.5% more than a developer without Resharper.
That gives you a basic financial model - if you get more than a 0.5% productivity gain, then it's worth it, if you get less, it isn't. Some corporates apply a minimum return on investment (ROI) factor - so if the factor is 1.2, then you would have to show a 0.7% benefit to get approval. The factor is very unlikely to be more than 3.
You could tweak that model - depreciate the license over 3 years, include the procurement costs, changing cost of capital, etc., but a simple, conservative model is likely to have the broadest appeal.
Then all you need is some evidence that you get more than a 0.5% productivity improvement. You could run a benchmark, or a pilot with a small number of developers for this. Pick some typical tasks and time them with and without Resharper. There is a 30 day trial version available so you could run a pilot before you have to purchase.
The PDF on the Resharper home page claims a 35% productivity increase - you can take that with a pinch of salt, but unless that's exaggerated by a factor of 70, it's still a worthwhile investment. The number of recommendations on the web, and developers claiming to buy it with their own money suggest that it isn't a wild exaggeration.
When you present the business case, you might like to illustrate that percentage as a dollar value too.
Developers only spend part of their day in their IDE, so you should probably adjust the expected returns downwards because of that. The real number is probably between 20% and 80%, but the lower end of the range might not be a politically acceptable number to present. You're interested in what proportion of the output is affected by the investment.
I don't have any connection with Jetbrains - and I'm answering a question about how to make a business case, not selling licenses! The anecdotal evidence from where I work is that the developers who have used Resharper have only good things to say about it. In some very specific cases it has saved weeks or months by automating mechanical tasks that have to be applied over a lot of files. The rest of the time it's hard to measure, but since the developers use it all of the time, they must be getting some real value out of it.
There's a quality argument too - you could measure this as a productivity increase, or a cost saving, or just an additional argument - depending on how quality issues are perceived at management level in your company.
Good luck with your business case.