Evidence of Jews, and Kingdoms of Israel and Judah in Israel:

Merneptah Stele - Dated to 1209/1208 BCE, "Canaan is captive with all woe. Ashkelon is conquered, Gezer seized, Yanoam made nonexistent; Israel is wasted, bare of seed."

Tel Dan Stele - Dated to 9th or 8th century BCE, mentions Yoram son of Ahab King of Israel, and Achazyahu son of Yoram King of the House of David, along with Yehu ruled over Israel. Note that the kingdom is acknowledged - in very high esteem - as the House of David, just as the northern kingdom is acknowledged as Israel. This correlates to 2 Kings.

Mesha Stele, or Moabite Stone - 9th century BCE inscription by Moabite king Mesha. Mentions Omri King of Israel, and though some dispute it, also mentions the House of David (Beit David).

Black Obelisk - Dated to 825 BCE by Shalmaneser III King of Assyria. Mentions and depicts King Yehu of Beit Omri (House of Omri/Israel)

The silver scrolls found by Gabriel Barkay, which feature the priestly benediction from Numbers, and a verse from Deuteronomy, is from the 7th century, prior to the Bavel exile. This is a particularly damaging find to all the documentary hypothesis fanatics.

Hezekiah's tunnel in Jerusalem is today as it is described being built in the time of Hezekiah in the Nach.

19 tumuli west of Jerusalem found from the time of the Judean monarchy.

Arad ostraca (#18 mentions the Temple in Jerusalem)

Azariah bulla (seal impression) found in 1978 during Yigal Shilohs excavation of old Jerusalem. The inscription consists of two lines of writing separated by two parallel lines. It reads Belonging to Azaryahu, son of Hilkiyahu. The impression does not mention the title of the owner. Yigal Shiloh, A Group of Hebrew Bullae From the City of David, Israel Exploration Journal 36 (1986), pp. 16-38

GBON () jar handles recovered from the Gibeon pool. Some inscribed "Hananiah" may have been associated with the person mentioned in Jeremiah 28:1. Other incised names on Gibeon jar handles: Amariah, Azariah, Domla, Geder, Hananiah, Neri, Shebuel.

Gemariah the son of Shaphan seal impression stamped on bulla. Found during Yigal Shiloh's excavations of Jerusalem in 1983, it probably belonged to the person recorded in Jeremiah 36:10.

Jehucal, the son of Shelemiah, the son of Shobi ( ) seal impression stamped on bulla. Found during Eilat Mazar's excavations of an alleged palace of King David in 2005, it probably belonged to the person recorded in Jeremiah 37:3 and 38:1.

Gath ostracon. Incised with nine letters representing two names ( ) etymologically related to Goliath ().

King of Assyria Sargon II's Conquest of Samaria inscription.

King of Assyria Tiglath-Pileser III's (745-727 BCE) inscription which mentions Yehoahaz of Judah, and claims to overwhelm Menachem, and place Hoshea as king over them.

Hanan's signet ring. Owned by a Paris collector, this valuable ring has been known to the scholarly world since 1984. The seals origin is unknown, but the shape of the letters indicate that it was used during the seventh century B.C.E. The seal is inscribed in three lines, each line separated by two parallel straight lines. The band is almost 1/10 of an inch in diameter, suggesting that it was designed for a mans finger. The inscription reads: Belonging to Hanan, son of Hilqiyahu, the priest.

Monolith by Shalmaneser III mentions 2,000 chariots, and 10,000 foot soldiers of Ahab the Israelite.

And I just stumbled accross a new story today about a cuniform tablet dating to 595 BCE referring to Nebo-Sarsekim, an official in the court of Nebucudnezzar King of Bavel. He is mentioned in Jeremiah 39.

The Yohoash inscription tablet is pretty much confirmed to be genuinely from the time of Yehoash. What is disputed by some is the inscription, however I don't believe it is disputed by all. It may have received a bad rap partially by being owned by the guy on trial for the James Ossuary.

There are some that are of unknown legitimacy as well. A stone seal with gold rim, attributed to King Manasseh of Judah. A pottery jug bearing an inscription claiming that it was given as a contribution to the Temple. Numerous bullae including ones which mention Biblical figures including King Hezekiah of Judah, the scribe Baruch and the prophet Isaiah.

Most interesting to me - though a small vocal minority is hesitant to admit it, and a large portion of people unaware - is we have two references to David in archaeology. But that is not what is most interesting to me. It is that the Kingdom of Judah is referred to in Steles by two different kings of two different nations - Aram and Moab - and they both refer to the kingdom as the House of David. The high esteem of his legacy and kingdom is thus far proven by the available archaeology, that they referred to his kingdom by his name.

Anyway, needless to say, there is more than enough legitimate proof of the existance of the Jews where the Tanach says they were, when they say they were. The House of David in the 9th and 8th centuries, and Israel's first mention in an Egyptian Stele in the 13th century. What we have are mentions of this large family by Egyptian, Aramean, Moabite, and numerous consecutive Assyrian Kings, plus much other archaeology.
I asked for strength, and G-d gave difficulties to make me strong. I asked for wisdom, and G-d gave me problems to solve. I asked for prosperity, and G-d gave me brawn and brain to work. I asked for courage, and G-d gave me dangers to overcome. I asked for love, and G-d gave me troubled people to help. I asked for favors, and G-d gave me opportunities. I received nothing I wanted. I received everything I needed. My prayers were answered.