I would like to know why the Romans, the emperor Honorius in this case, didn't use the Roman Army (i.e. Legion). According to the Notitia Dignitatum, there were still many Legions around the Western Roman empire that could have been used to defeat the Alaric's Goths. However, Honorius had extended and unsuccessful negotiations with Alaric at the point of thinking to flee Ravenna. This scenario ultimately led to the sack of Rome, but the question is, could this have been avoided by using the army? If that's the case, why didn't the Romans do so? This question seems to be duplicated with (Where were the western legions when the Roman empire fell?) but I am pointing exactly to the siege of Rome of 410 whereas the other topic addresses the "Fall" of the Roman Empire.
The Western Empire did not use the army because they did not have an army to speak of.
In 406, the Vandals passed over the frozen Rhine and rampaged through Gaul. Imperial forces in Britain joined a usurper and there was a three way combat under way. No help there.
The Imperial Generalissmo, Stilicho, was working on and off with Alaric to regain the parts of the Balkans ceded to the Eastern Empire after Adrianople. These were good recruiting grounds for troops. This made little progress. At other times they fought, but Stilicho was not willing to destroy Alaric's force as he saw them as a possible ally.
Stilicho could not intervene in Gaul since he had had to face the invasion of Italy by Radagaisus the Goth in that same year. He managed to capture the force, and incorporated the troops into his army. Their families were settled into Italian towns.
Stilicho next was faced with a crisis, as the Eastern Emperor died. He wanted to intervene in the secession to gain his goals in the Balkans. This gave his enemies in court at Ravenna the chance to overthrow him and he was executed. In the aftermath, the citizens in the towns rioted and killed the families of the Gothic soldiers in the Army. These soldiers, virtually the entire army, deserted in a body and joined Alaric.
Alaric, without his backer Stilicho, tried to gain his goal of a homeland and power in the Empire by a direct confrontation with the now virtually armyless Empire. However, the total lack of siegecraft allowed Honorius to ignore him, aside from sending out the odd assassin. Alaric was baffled until some unknown person let the Goths into Rome.
The situation of Rome at that time was entirely different than what you imagine. The first key thing to understand is that the western empire and eastern empire were split on political, ethnic and religious lines. The main power in the world (until 408) was Theodosius, the emperor in Constantinople. Constantinople was founded by the first Christian emperor, Constantine, and was thoroughly Christian and had a stable government. The emperors in Constantinople, including Theodosius, had an antagonistic relationship with Rome, which was half Christian, half pagan. In the 4th century Rome became increasingly chaotic and ungovernable with large mobs of poor people, wealthy decadent pagan Romans, foreign soldiers and other strange elements in a huge seething mass. Out of this chaos, an unknown named Eugenius was put forward as the western "emperor" in 392, defying the rule of Theodosius. The various Frankish barbarians then holding sway in Rome, who were pagans, had sponsored this revolt. Theodosius duly took an army and crushed them in battle in 394, installing his juvenile son, Honorius, as the titular ruler of Rome. Real power was wielded by the Vandal general Stilicho who was the head of the foederati, the German mercenaries loyal to the Roman empire. Note that these were Christian barbarians, unlike the Frankish barbarians in Rome who were pagans.
Stilicho gathered the remnants of the government who still supported Constantinople and moved them along with Honorius to Ravenna, a large military base which had a good port. There, they set up a rump government of the west which had very little control of Rome and absolutely no control of the rest of the western empire (Gaul, Iberia, Britain, etc).
This precarious status quo held for 14 years, but when Theodosius' son Arcadius died in 408 everything collapsed. The fawning courtiers surrounding Honorius intrigued against Stilicho and had him murdered. Honorius himself was useless, a spoiled 26-year-old brat. Pagan Romans and westerners took this as a signal to overthrow the now-leaderless foederati who were Constantinople's only forces in Italy. They went around massacring foederati and their families. The remaining foederati then fled to Alaric, the Visigoth, who was a Christian and loosely aligned with Constantinople. Alaric, now with the full support of the Italian foederati (Constantinople's warriors) and the moral high ground, descended on pagan Rome and sacked it.
As far as Honorius was concerned the pagan Romans, especially those in the countryside, were now getting what they deserved. The western pagan/Frankish legions in Gaul probably sympathized with their allies in Rome, but they were too weak, disunified and far away to do anything about it.