You can trip a prone opponent, but it's pointless.
First, let's look at the basic case. Trip is an action you can take against an opponent as a melee attack. The only requirement is that the opponent is no more than one size category larger than you. There is no requirement that the target be standing.
However, since the only effect of being tripped is becoming prone, successfully tripping a prone target will have no additional effect*.
Now let's consider the feats.
If you deal 10 damage in melee combat, you can make a trip attack as a free action as per Knock-Down. However, Improved Trip says:
If you trip an opponent in melee combat, you immediately get a melee attack against that opponent as if you hadn’t used your attack for the trip attempt.
This implies that the game expects Improved Trip to occur only when you use it in place of an attack. If you make an trip attack as a free action using Knock-Down, you won't get the extra melee attack from Improved Trip.
As Hey I Can Chan mentions, this interaction is expressly forbidden in the errata for Knock-Down:
Use of this feat cannot be combined with Improved Trip to generate an extra attack
Finally, let's consider the practicality of generating infinite free actions. The rules say:
there may be limits to the number of free actions you can perform in a turn.
It's within the GM's power to prevent additional free actions beyond what could reasonably be possible within a combat round.
*There may be exceptions, such as a terrain that damages creatures that fall prone, but falling prone implies they were not previously prone. A GM may allow the damage anyway (representing you flinging or pushing the target around on the damaging terrain).