What is the proper response to Ariel Sharon's illness? The Israeli Chief Rabinate has asked all Jews regardless of affilliation to say tehillim (psalms) for General Sharon, specifically chapters 6, 20, 121, 130 & 142. (But any chapter[s] will do just fine; say what you can.) The entire Book of Tehillim is online in both Hebrew and English.
Studying Torah is also important. Online versions of the Babylonian Talmud can be found in Hebrew with other extensive study aids in English, and in English translation without notes or study aids. The Kitzur Shulkhan Arukh ([very] abridged code of Jewish law) is online in a bilingual edition. Rabbi Chaim Jachter has extensive essays (similar to responsa) on various aspects of practical Jewish law online in English. And, last but not least, the entire Tanakh (Hebrew Bible) is online in Hebrew and in English.
It is also customary to give tzedaka (charity). Any amount is fine. Do what you can.
Doing good deeds is also important. Perhaps you can find an elderly person or a shut-in to visit. Certainly there are many people who need assistance in myriads of ways – find one that is right for you and do it.
A simple heartfelt prayer offered in any language, either within the Refaenu blessing of the Amidah prayer or as a stand-alone at any time, is very appropriate.
May God grant Ariel Sharon a full and speedy recovery.