You can make an auction for the private key of 34dqaqvQNWMgbMJmmxVa8LeGz7St6ATT97 addy...
he has already done that in a much better way which is to place the reward as a high transaction fee. the 0.27 bitcoin is currently worth nearly $2500 which he could increase in the same manner (bumping the fee).
if a miner wanted to accept the offer they could just include the transaction in their next block they mine and claim its fee. others that aren't miners can't do anything about it.
Well this depends on how you look at it.
There can also be a problem as follows, suppose i manage to push this tx into the network, then it will be mined by some unknown miner, or pool, which will get a high .27 fee, then he will be happy, and the OP will be happy to see that he regained access to his funds, but i will be here empty handed.