According to the first answer of This question parts of Halley's comet hit the Earth every spring as part of the Eta Aquarid meteor shower. If this is true, then how is the comet still in one piece? If parts of this thing are breaking off every year as it travels through space, that means the comet must be getting smaller. If I am not mistaken, a comet can only be so big. This question addresses just how, in all the time Halley's comet has come back before, and broken parts off of itself each time, can it still be in one piece.
The basic answer to your question though is that Halley's comet is unlikely to be very old (in the sense that it has not spent a large amount of time orbiting the Sun as closely as it does now). Most comets are perturbed into their current orbits from way beyond the orbit of Pluto and then live comparatively brief lives as they gradually disintegrate.
According to this NASA source the comet should lose about 1-3 metres depth of surface every time it swings around the Sun. Let's say it started off with a 10 km radius; this would mean it could complete 3,000-10,000 orbits before vanishing. Since the orbital period is 76 years, this implies a lifetime of 228,000 to 760,000 years. Thus if Halley's comet has been in its current orbit for less time than this then there is no mystery.
An interesting calculation is presented by Hughes (1985) who estimates the current mass of the comet, its estimated mass-loss per orbit around the Sun, and then compares that with the estimated total mass in the meteor stream you mention. The conclusion is that the comet has made about 2300 previous passages around the Sun and so is only about 175,000 years old (in its current orbit) and will probably survive for another 2300 orbits.
Comets can be fairly large initially, multiple kilometers across. However, just as parrots become ex-parrots, comets eventually become ex-comets. They shed volatiles, sometimes shed pieces, and yet other times even disintegrate. Some of the things we think of as asteroids might well be comets that have lost their volatiles, leaving just the non-volatile (e.g. rocky and dusty) material behind.
Comets are extremely volatile because mostly they consist of ice and dust, once they reach the perihelion i.e closest point to the Sun in an orbit around the Sun they sublimate forming the Coma and the tails this is also the reason meteor showers out of comets happen, also whenever a comet reaches some celestial object the Tidal forces of the object melt some pieces of the comet, and Halley's comet's orbit is in a short-term orbit (because it's orbit was perturbed due to the Solar system being chaotic) so it comes close to the Sun frequently. So in reality it is actually reducing in mass every time it comes at perihelion
But however it's rate of loss of mass is slow compared to it's entire mass. So recent research suggest that Halley's comet might evaporate in a few thousand years.