Basically, you do need to connect to a mining pool---which you can provide yourself by running the bitcoin client if it indeed has to be solo mining for you. Note that from your question it is obvious you do not understand the implication: Solo mining can only get you at least a block reward (or nothing), so currently, you can never get just a single Bitcoin, you have to accept at least 25 in solo mining. Only by joining a mining pool can you get smaller rewards for less difficult-to-find hashes.
Anyways, connecting can be as simple as doing a RPC call. With this, I guess I can refer you on to the bitcoin.stackexchange question "What is the standard protocol for pools/miners?".
Here's the summary, slightly updated from the linked question:
The legacy bitcoin RPC call you could use is getwork. You can probably find a mining pool that supports it (but I haven't checked that). One minimalistic sample implementation, Minimal-Bitcoin-Miner on github, can be found linked to and with what you call the "tryHash" bit discussed in the question linked to by RentFree in a comment to your question.
The newer getblocktemplate may work better for you. Many mining pools seem to support it.
If you want to work in a specific pool, check if they are inventing their own protocol, just in case that suits you even better. Slush's protocol is one example.
Good luck and let us know when (if ever) you can unveil your hash-solving black box!
If it promises to work well enough to support paid help, may I apply, having expertise spanning pretty much anything that might be relevant (quantum physics, electronics, programming, mathematics)?