So essentially the company doubled in value - the investers thought it was undervalued and were willing to pay the difference

No - the investors were willing to add $169.5M of value to the company by giving it $169.5M in cash.

The company is "worth" the same as it was before, plus $169.5M new cash in the bank.

In return, the investors received newly-issued shares at a price based in the pre-investment valuation of the company.

So there's no paying any "difference"; only increasing the valuation of the company by injecting new cash into it and creating new shares.