You don't submit just the nonce. You submit the entire block (including all transactions, and a valid block header including the nonce) to the nodes you are connected to and they relay it to nodes they are connected to and eventually every node in the network receives the block. You get paid because the block you submit has a coinbase tx which creates 50 BTC (currently) from nothing and sends it to the reward address you provide.
Still solving with paper is just beyond pointless.
Current difficulty is 2,694,047. That means on average it will take 2,694,047 nonce ranges to solve a block.
Nonce range is 2^32 so it will take on average 11,570,843,758,886,900 solved hashes to find one which meets the target requirement (determined by difficulty).
If you could solve perform the SHA-256 double hash by hand every minute (unlikely) it would take on average 22 billion years to solve a block.
As a god, I originated from the big bang. 50% chance @ 13 billion years have been sufficient. (obviously i'm not doing it on paper, I just want to understand, and create, an interface)
So how do you connect to the nodes? I assume that which one is arbitrary, but what is the 'send' standard? Is it a post, or a get, JSON format like the output? how do you get the node to access your data?