What likely happened is you banged the lid or sat something heavy on it. The electrical connections between the LCD glass and the driver logic on the back broke.
These are hard displays to replace, I would recommend you replace the complete lid assembly.
As you didn't give us your exact model info I can't point you to the IFIXIT guide or the needed part. If you do we can offer that to you.
As to why your HDMI connection to your TV failed you might want to check which HDMI port you used as well as what you need to set within the TV to activate the port. The other possibility here is the mini DisplayPort to HDMI adapter you got is bad.
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=== Update ===
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Here's the IFIXIT guide for your system: [https://www.ifixit.com/Guide/MacBook+Air+13-Inch+Mid+2011+Display+Assembly+Replacement/6361|MacBook Air 13" Mid 2011 Display Assembly Replacement] and here is the full assembly: [https://www.ifixit.com/MacBook-Parts/MacBook-Air-13-Inch-Late-2010-Mid-2011-Display-Assembly/IF188-050|MacBook Air 13" (Late 2010/Mid 2011) Display Assembly]
ZZZ is correct here your problem is the display.
What likely happened is you banged the lid or sat something heavy on it. The electrical connections between the LCD glass and the driver logic on the back broke.
These are hard displays to replace, I would recommend you replace the complete lid assembly.
As you didn't give us your exact model info I can't point you to the IFIXIT guide or the needed part. If you do we can offer that to you.
As to why your HDMI connection to your TV failed you might want to check which HDMI port you used as well as what you need to set within the TV to activate the port. The other possibility here is the mini DisplayPort to HDMI adapter you got is bad.