I am wondering if it is possible to write Solidity code to Mine Bitcoins on Ethereum's distributed computing platform?
In theory it's just hashing so yes, you could make a contract that did Bitcoin mining. In practice it would be impossibly expensive, and you'd never find a block.
Ethereum's distributed computing platform is useful for when you want to do a single shared calculation, and have lots of people verify and share it and agree on the result. This makes it extremely expensive to do even trivial calculations. Mining is the opposite: You want lots of people to do independent calculations, and only share the occasional winning result.
There may be some useful ways to use Ethereum for organizing mining pools, however. It may also be interesting to use Ethereum to incentivize certain behaviour that you want; For example, you might want to make a bounty for a miner who signaled for or against a Bitcoin protocol change you wanted, or who orphaned blocks containing a particular type of transaction that you thought was undesirable.