"As I opened my eyes, a bright light flashed on, and forced me to close them again. After a few seconds, I dared to force my eyelids open once more, and slowly began to take in my surroundings."
"I dared to force my eyelids open once more.." My issue here is with the use of "dared". Dare is used to describe an action that one isn't likely to do, or is an action that is very risky/scary/treacherous. Opening your eyes after sleeping and closing them due to bright light, is a very natural thing that everyone experiences. Dare comes off as too powerful and I just don't feel like it fits well. Dare also gave birth to "force my eyelids open", but really, its another natural action we all experience. You were giving you eyes a chance to adjust to the light. If it was forced, then you would have had to add that you were fighting through the pain that your eyes were experiencing from the shock of going from extreme lack of light to extreme brightness of light. Because everyone can relate to this part, you shouldn't put too much emphasis and power in it or you come off as maybe an exaggerator for the sake of telling a story to appease your audience.
"Using my powerful senses of deduction, I concluded that I was in some sort of hospital room"
My issue with this part is for a lot of the same reasons of the part above. "Using my powerful senses of deduction.." You come off a joker, an exaggerator, or just someone that wanted to find a way to incorporate the word deduction to show off that you know it, and thus have a good vocabulary. The last part is wise to do, but incorporate the word more casually. The way you say "powerful senses of deduction" for something as simple and obvious as the detection of being in some sort of hospital room, does not fit.
The moment you said "tubes in my arm", I knew confidently you had to be in a hospital, for which I suspected after "some sort of white room". You then give even more hints that there is not anyone that wouldn't know you were in a hospital room.
So to say this is "powerful sense of deduction" is really just a common sense of deduction. I would assess this as a lack of understanding or maybe a lack of judgement, where you will over-compliment or over-insult things. Shows a sense of inflation, or exaggeration.
I don't actually think this of you based on the rest of the writing, I think you were just wanting to really write something good, but put a couple too many bells on.
Conclusion: I think its well written. I think you achieved the goal of the prompt and don't come off as fake. There is a slight sense of you wanting to get pity by speaking about losing your mother in an accident, but I think there is enough that over-weighs that fact and makes it more about something that is central to your identity. Yet, this makes it more important to watch out for exaggerations.
That is just my opinion though. I have no idea who the person/people are to read this essay, and so they may have a totally different idea. So, do as you like with my opinion and don't be afraid to tell me what you think of it.