Why don't planets give off their own light?
Why don't the planets glow like stars?
Where did you get those numbers regarding Mercury's composition? They aren't accurate; the rocky planets in our solar system aren't composed of that percentage of hydrogen and helium.
All matter radiates (except if it's at absolute zero temperature), regardless of its composition (you got that of Mercury badly wrong). The most important form of radiation is the black-body radiation which only depends on the temperature of the material, but line emission and absorption may also be important (but depends on the composition and ionisation state of the material) and other emission and absorption processes.
Stars are hot enough (the Sun has surface temperator $\sim5700$K) for the black-body radiation to peak in the visible part of the electro-magnetic spectrum. As a star shines, it looses energy, i.e. it cools, reducing the gas pressure that stabelizes it against gravitational collapse. In stars, this energy loss is balanced by the energy production from thermonuclear fusion in the core (requiring temperatures $\sim10^9$K). The transport of this energy to the surface makes stars non-trivial.
In planets and brown dwarves, there is (by definition) no thermonuclear energy source. Therefore, these objects must shrink, which generates energy from gravity. However, they cannot shrink indefinitely, as ultimately quantum mechanics becomes important: Pauli's exclusion principle demands that the electrons cannot be arbitrarily closely packed. Thus, for brown dwarves and giant gas planets (but also white dwarves) further shrinking is halted at a radius comparable to that of Jupiter (Jupiter itself is still shrinking at a very small rate). These as well as all smaller objects then merely cool down very much like a piece of glowing coal.
The situation is a often more complicated by sources of energy. Planets, for example are
irradiated by their host star, which may dominate the energy gains at their surface (in addition, the Earth gains energy from nuclear fission in its core). The balance between this energy gain and the loss by black-body radiation determines the temperature of a planet.
The Earth, for example, radiates in the infrared. However, the radiation losses are also regulated by line absorption of that infrared in the higher atomsphere by so-called green-house gases, in particular CO$_2$.
Related Q: Why does Jupiter emit more energy than it receives? http://physics.stackexchange.com/questions/25417/why-does-jupiter-emit-more-energy-than-it-receives "[Jupiter] is still contracting at a rate of ~3 cm per year while its interior cools by ~1 K per million year."
But what prevents the gas giants from compressing further and starting thermonuclear fusion?
@Yashbhatt They lack enough mass to pack hydrogen to the density required for fusion.
@WayfaringStranger Yes. Just like the stars. But if Jupiter is contracting as you mentioned, then does it mean that it will possibly start fusion or is it's mass below the critical mass to start fusion?
Stars' light is produced by thermonuclear fusion of hydrogen, helium and other elements at their nucleus. This processes are impossible in planets since you need high pressures and temperatures >10,000 k
See my answer. The energy source of a stars is thermonuclear fusion in their core (requiring $T\gg10,000$K), but the light is emitted at the stellar surface as the black-body radiation with $T\sim5000$K.