How can I use asymmetric encryption, such as RSA, to encrypt an arbitrary length of plaintext? RSA is not designed to be used on long blocks of plaintext like a block cipher, but I need to use it to send a large (encrypted) message.

How can I do this?
The solution to this problem is to use hybrid encryption. Namely, this involves using RSA to asymmetrically encrypt a symmetric key.

Randomly generate a symmetric encryption (say AES) key and encrypt the plaintext message with it. Then, encrypt the symmetric key with RSA. Transmit both the symmetrically encrypted text as well as the asymmetrically encrypted symmetric key.

The receiver can then decrypt the RSA block, which will yield the symmetric key, allowing the symmetrically encrypted text to be decrypted.

This can be shown more formally as the following. Let $M$ be the plaintext, $K_{AES}$ be the randomly chosen AES key, and $K_{Pu}$ be the receiver's public RSA key you already have.

$$ C_1 = E_{AES}(M, K_{AES}) $$
  $$ C_2 = E_{RSA}(K_{AES}, K_{Pu}) $$

Then, send both $C_1$ and $C_2$.

Let $K_{Pr}$ be the receiver's private RSA key. The receiver can then recover $M$ as

$$ K_{AES} = D_{RSA}(C_2, K_{Pr}) $$
  $$ M = D_{AES}(C_1, K_{AES}) $$

(To allow streaming decryption or large messages, you would usually send $C_2$ first and then (the much larger) $C_1$.)