Quite a problem. First of all, much greater care must be taken. I work with circuits using SMD components a lot, and although they are delicate, you've got to be relatively heavy-handed to remove them from a PCB. I say relatively, because you become used to treating PCBs populated with them extremely gently and carefully.
Now, the component could be a resistor, an inductor or a capacitor. If you're lucky, it's just a decoupling capacitor and it's loss may not present any noticeable problem. You will be unable to discover the part's value even if you discover what part it actually is - that's the problem.
If you still have the part, you could try to solder it back on, but it's likely that by removing it, you've also torn away the copper pads beneath it. From the image it's hard to tell if there are any tracks going to it that you could solder to if the pads are gone, or if it uses through-holes to inner layers. You'd need a very fine tip soldering iron - preferably a temp-controlled soldering station, and fine solder (I use 0.3mm). But you'd find it almost impossible, and the component will keep sticking to the iron's tip unless you're careful and use tweezers to hold it in place while dabbing one end with solder to temporarily hold it.
Personally, I use solder paste applied through a very tiny needle from a syringe and a hot-air soldering station for this kind of job. Maybe there is a TV repair place near you that has such equipment and would solder the part in place for you?
If you've lost the part, then try the device without it and see what happens. If it doesn't work properly, then perhaps you could buy a faulty device or logic board cheaply and scavenge it for necessary parts.
I've spent many years soldering (family trade is electronics repair), and a long time developing my skills at fine soldering SMD parts and using solder paste and a hot air station. I'm not sure how someone would fare with this kind of job without the right skill level and tools...
However, best of luck with it and let us know how you get on.