Might it be Dragon Fall by Katie MacAlister?
You Flirt With Fire... For Aoife Dakar, seeing is believing-and she's seen some extraordinary things. It's too bad no one else believes that she witnessed a supernatural murder at an outdoor fair. Returning to the scene for proof, Aoife encounters a wise-cracking demon dog-and a gloriously naked man who can shift into a dragon and kiss like a god. Now thrust into a fantastical world that's both exhilarating and terrifying, Aoife is about to learn just how hot a dragon's fire burns. When You Date a Dragon Kostya has no time for a human woman with endless questions, no matter how gorgeous or tempting she is. He must break the curse that has splintered the dragon clans before more of his kind die. But his powerful attraction to Aoife runs much deeper than the physical-and there may be more to her than even his sharp dragon eyes can see. To survive the coming battle for the fate of his race, he needs a mate of true heart and soul . . .
From this review:
The storyline at first made me a little cranky. The heroine, Aiofe, witnesses a crime that is completely unexplainable. No one believes her story, so she is dumped in a medical facility so she can see the errors of her ways. So on behalf of her, I already had my back up. She is eventually released after suffering some horrible treatments, but when she returns home, the sister (cow), who put her in the facility, leaves her immediately. So much for caring for her mental well-being. Within hours, Aiofe's world is changed forever.
The hero of the story, the leader of the black wyvern, Kostya, has a permanent frown on his face. He's been through a lot in recent years, which has made him weary. When he meets the heroine he knows what she is to him but denies it till the end.
There's this ring, sort of like Lord of the Rings, but different. Everyone wants it, but doesn't necessarily want to possess it. This ring holds a lot of power and tends to create mischief. This story mainly focuses on Aiofe learning and harnessing its power.
Found with search terms of site:goodreads.com woman insane asylum dragons
You can read an excerpt here.
Dr. Barlind’s note: the following is the interview held at the admission of Patient A upon the demand of her family members. Clear signs of schizophrenia were demonstrated, and a reluctance by Patient A to admit that her story more resembled a fantastical movie than real life. She expressed a great desire to tell her story, however, which was encouraged, and which we hope will facilitate recovery.
INTERVIEW BEGINS
Dr. Barlind: Good afternoon, Miss A. How are you feeling?
Patient A: I’ve been better, and my name is Aoife, not Miss A. It’s Irish, and pronounced EE-fuh.
Dr. Barlind: My apologies. Aoife. Some patients chose to be anonymous in our reports, but I will make a note of your preference. Would you like to tell us what happened that made your brother and sister decide you needed our care?
Patient A (shuddering): I’d rather not think about it, but I suppose if anyone is going to do anything about it, then I’ll have to tell you what happened last night. It was last night wasn’t it?
Patient exhibited signs of distress, and was reassured that the triggering event had occurred the past evening.
Patient A: OK, good. I thought I’d lost some time there, too, which let me tell you, isn’t as freaky as it sounds. Where should I start?
Dr. Barlind: Where you are comfortable beginning?
Patient A: I guess it all started with the date. I had no idea that anything…weird…was going to happen. I mean, Terrin looked perfectly normal. He certainly didn’t seem like the type of man who could die and resurrect himself at will.
Patient A shuddered again, and rubbed her eyes as if wishing to remove mental images, but ceased before self-harming.
Dr. Barlind: Why don’t you start with your date with this man, then.
Patient A: Yeah. The date. It started all right. Nothing fantastic, but pleasant enough…
