Let’s break it down:
1. 5d6 cold damage.

This is roughly equivalent to a 5th level caster’s Shocking Grasp with the benefit of increasing a melee touch attack to a ranged one. As a single use, use activated item this would be in itself worth 250gp although you’d probably want to add something for the added range. Definitely no more than double the price as it’s 3d6 short of Scorching Ray (2nd level) at 5th level. Lets say 300-350gp.

2. DC20 Fort save to negate perpetual unconsciousness which requires, at best, a 4th level spell (if you happen to have a Bard on call otherwise 5th level) to undo.

The effect is roughly equivalent to a save or die spell such as Phantasmal Killer (4th level) which requires the 5th level Raise Dead to undo and has a similar dual saving roll (Will save + Fort save vs touch attack + Fort save) but the DC seems a tad on the high side (CL 11th => 6th level spell => Int/Wis/Cha 16 => DC10 + 6 + 3 = 19). Additionally, the effect is untyped unlike P.K. which is both a mind-affecting and a fear effect. The only disadvantage over P.K. is the range: ~close with increment penalties vs medium. This is more than balanced by the lack of typing.

Anyway, comparisons aside, basing this effect purely on CL 11th, 6th level spell, single use, use activated item this in itself gives a price of 3,300gp.

Costing (assuming multiple different abilities):
3,300 + ((300 to 350) × 1.5) = 3,750 to 3,825gp

I’d be tempted to round that up to 4,000gp.

I’d also recommend typing it as a cold effect and giving bonuses on the saving throws for those with resistances to cold (+1 for every 5 points resistance sounds reasonable) and those with immunity to cold being (*GASP*) immune. The effect should also be fatal to creatures which are vulnerable to cold (i.e. Fire subtypes). Additionally, the DC should come down one or increasing the CL to 13th, the base spell level to 7th and raising the price accordingly to ~5,000gp.

Good to see a single-use artifact for a change although I’d expect such a thing to be a ‘random’ find rather than an actual objective (“All that effort for something we can only use once?”) or loot from a confrontation with a major antagonist (who, given its potential in combat, would use it on the PCs). Unless of course said person were stockpiling it in an effort to resurrect the genie or to hold the world to ransom and fulfill their dreams of becoming a Bond villain…