@besides the excellent answer provided by my colleague @teach_ni is a great verification.
If you can't see your screen and if you are sure it has failed, you will have to replace teh complete display. The pieces like the LCD (AMOLED), touchscreen and glass are all sandwiched together. Now, there are ways to separate those, but that is mostly done for a Glass only repair and require special tools and skills. The tools alone will cost more than a replacement display. Something where the screen still shows but the front is cracked.
In your case it's the actual display which has failed. The only repair for that is the complete display replacement. It is the quality of those displays that makes them so expensive. These are not your common LCD screens but are definitely a step up from those. Sorry to say this but the price you were quoted is actually a pretty good price