Short answer:

The 5,000 number is from around 1897. Prior to that point, the Korean armed forces had been much larger, but there'd been several events that'd derailed things.

There were serious pay shortages and poor serving conditions in the traditional army. The Joseon government had also been purging the army of older or unfit soldiers, which shrank the size of the army (at least 1,000 men were pushed out). These events meant the army's size shrank. They also created resentment among soldiers that lead to the 1882 Imo Mutiny.

The mutineers turned on a Japanese-trained "model" army unit called the Special Skill Force. The Special Skills Force had been intended to serve as the nucleus of a modernized Korean army. As a result, it was given better pay and better treatment, exciting jealousy among the mutineers. The mutineers killed many of the Special Skill Force's Japanese advisers and the Korean officials who had supported its creation This understandably stalled Korean military modernization and expansion.

Korea in the late 1800s was extremely chaotic politically, poor economically, and disorganized administratively. A large-scale expansion of the military just wasn't feasible given the poverty and lack of political will in the country for such an effort.

It wasn't until the Gwangmu Reforms of the late 1800s and early 1900s that another serious attempt at military modernization was made. But by then, it was too little, too late. Japan was too strong militarily and diplomatically, and they were able to forcibly disband the Korean armed forces in 1907.

Longer answer: We need to break this time period (the late 1800s) into two periods: the last days of the Kingdom of Joseon, or Dae Joseonguk (1392–1897) and the short-lived Korean Empire, or Daehan Jeguk (1897–1910).

The Gwangmu Reforms of roughly 1897 to 1907 take place during the Korean Empire, as part of the Empire's efforts to strengthen and modernize Korea in response to growing Japanese encroachment. To understand the civil, political, and military reforms of the Korean Empire, we need to rewind the clock a bit to the last days of Joseon.

In the late Joseon period, Japan wasn't the only threat that the Joseon government faced. For example, Western powers like the United States and France had ambitions of their own in Korea.

To confront these outside forces, the Joseon kingdom did have a decent-sized army. I don't have exact numbers, but an estimate of over 100,000 men nationwide seems reasonable. Most of the army was stationed in the capital, Hangang (now Seoul) or stationed in coastal or border defenses on the periphery of the country. Of course, this army was reliant on swords, spears, smoothbore cannon, and matchlock muskets, which left it generations behind the modernized forces threatening it.

In the 1860s and 1870s, Joseon was under pressure from outsiders to open up their country to traders and missionaries. Xenophobic and traditionalist forced within the Joseon court strongly resisted Western overtures. French Catholic missionaries were seen as a threat to the religious and moral authority of the Joseon court. Dozens of French missionaries and Korean converts were executed, giving the French Far East Squadron casus belli for a punitive expedition in 1866. French troops successfully stormed Joseon forts on the western coast of Korea, but the small expedition wasn't able to take the capital at Hangang (today's Seoul) against the larger Korean forces. The Koreans had 10,000 men in the area, enough to prevent a French advance further inland.

That same year, the American merchant ship, General Sherman had been burned and its crew massacred after its trading expedition had descended into kidnapping.

In 1871, the Americans sent their own mission to Korea to "open up" Korea for trade, much as Commodore Perry's 1853 expedition to Japan had pressured Japanese leaders to acquiesce to a trade deal. The American fleet also hoped to investigate the disappearance of American ships like the General Sherman.

The American Expedition of 1871 followed a similar course to the 1866 French campaign. Well-armed, well-disciplined, and well-led American marines and bluejackets took several Korean coastal forts and inflict lop-sided defeats on the Korean defenders. Armed with matchlock muskets, swords, and old cannon, Korean troops didn't have much chance against Americans with rifles, bayonets, and exploding shells. In one assault, the American lost just three dead, while killing over 240 Korean defenders.

However, there just weren't enough Americans to decide the issue, so the American expedition had to weigh anchor and head for home, without having managed to "open up" Korea.

In 1875, Japan engaged in a similar round of "gunboat diplomacy" during the Gangwha Island incident of 1875. A Japanese gunboat shelled a Korean coastal fort and stormed it. Japanese troops armed with Western-style rifles made short work of the outgunned Korean defenders. This incident, and other Japanese pressures eventually lead to the Japan–Korea Treaty of 1876, which finally opened Joseon trade from the Japan and other powers.