There were two battles in Lexington. The first one is listed as taking place Sept. 13-19.
Here's a summary from the Lexington Area Chamber of Commerce website:
Lexington was the site of two of the largest battles in the western
campaign of the Civil War. The first and most famous, known as the Battle of
Lexington is better known as the Battle of the Hemp Bales. On September 12th,
1861, somewhere between six and ten thousand Missouri National Guardsmen were
led by Major General Sterling Price. Price began a siege against the Federal
military post positioned in the old Masonic College. They were Commanded by
Colonel James A. Mulligan. Price's army mounted an assault on September 18th.
Some of Price's army used hemp bales as moving breastworks while they moved up
the river bluffs and closed in on Mulligan's headquarters. On the third day of
the siege, Mulligan's troops surrendered. The combined ed casualties numbered 73
dead and 270 wounded. The battlefield remains today on the bluffs of the river
in virtually pristine condition and stands as a state park. In an attempt to
crush General Price's headquarters located on Main Street, Mulligan's troops
fired cannonballs from the battlefield. One such cannonball missed the mark and
became lodged in the leftmost pillar of the Courthouse where it remains to this
day. That cannonball has become an iconic symbol for Lexington.
Following the victory at Wilson’s Creek, the Confederate Missouri State
Guard, having consolidated forces in the northern and central part of the state,
marched, under the command of Maj. Gen. Sterling Price, on Lexington. Col. James
A. Mulligan commanded the entrenched Union garrison of about 3,500 men. Price’s
men first encountered Union skirmishers on September 13 south of town and pushed
them back into the fortifications. Price, having bottled the Union troops up in
Lexington, decided to await his ammunition wagons, other supplies, and
reinforcements before assaulting the fortifications. By the 18th, Price was
ready and ordered an assault. The Missouri State Guard moved forward amidst
heavy Union artillery fire and pushed the enemy back into their inner works. On
the 19th, the Rebels consolidated their positions, kept the Yankees under heavy
artillery fire and prepared for the final attack. Early on the morning of the
20th, Price’s men advanced behind mobile breastworks, made of hemp, close enough
to take the Union works at the Anderson House in a final rush.
From the Wikipedia: First Battle of Lexington:
On the evening of September 19, soldiers of Brig. Gen. Thomas A. Harris's 2nd Division (State Guard) began using hemp bales seized from nearby warehouses to construct a moveable breastwork facing the Union entrenchment. These bales were all soaked in river water overnight, to render them impervious to any heated rounds fired from the Federal guns. Harris's plan was for his troops to roll the bales up the hill the following day, using them for cover as they advanced close enough to the Union garrison for a final charge. The hemp bale line started in the vicinity of the Anderson house, extending north along the hillside for about 200 yards. In many places the hemp bales were stacked two high to provide additional protection.
Early on the morning of September 20, Harris's men advanced behind his mobile breastworks. As the fighting progressed, State Guardsmen from other divisions joined Harris's men behind the hemp bales, increasing the amount of fire directed toward the Union garrison. Although the Union defenders poured red-hot cannon shot into the advancing bales, their soaking in the Missouri River the previous night had given them the desired immunity to the Federal shells. By early afternoon, the rolling fortification had advanced close enough for the Southerners to take the Union works in a final rush. Mulligan requested surrender terms after noon, and by 2:00 p.m. his men had vacated their trenches and stacked their arms.
Many years later, in his book The Rise and Fall of the Confederate Government, Southern president Jefferson Davis opined that "The expedient of the bales of hemp was a brilliant conception, not unlike that which made Tarik, the Saracen warrior, immortal, and gave his name to the northern pillar of Hercules."
There's a page about the state historic site that doesn't include a whole lot of information, but you should check it if you think you might like to visit Lexington.