How did the invasions of the mongols weakned the roman empire?

The Huns were not successful invaders of the Roman Empire. They ravaged the eastern part of the empire, but withdrew when they exacted a tribute; they were not interested in invading. Attila then tried to invade Gaul, but he was repelled by a combined force of Romans, Franks, the Burgundians and Visigoths in 451. In 452 he started an invasion of Italy, but had to give up because of famine in Italy and because a Roman attack on the Hun heartland.

The successful invasions were those of the Vandals, Sueves and Alans, who invaded Gaul in 406 and moved to Spain in 409. The Vandals and Alans then took over north-western Africa, where they established the Kingdom of the Vandals. This invasion and other events was what led to the fall of the western part of the Roman Empire. The eastern part was not affected and continued to exist for nearly 1,000 years. Prior to that, the Romans had been able to repel all attempts at invasion for more than 200 years.

It was not so much the attempted invasions which could act as a gauge for the military strength. The Romans struggled because there were continuously being attacked by many peoples along the frontiers of the rivers Rhine and Danube. Thus, the attempted invasions were more of an indicator of how difficult it was to defend thousands of miles of borders in an area where there were peoples who were migrating southwards towards these frontiers. Moreover, the Romans were engaged in continuous wars with the mighty Persian Empire. Thus the Romans were overstretched.

What was more of an indicator of diminishing Roman military power was an internal one. They increasingly had to rely on Germanic allies to support their troops and on hiring mercenaries and enlisting soldiers form the Germanic peoples. Even the Huns had provided them with mercenaries. The size of the empire and of the military requirements for its defence from continuous attacks meant that the Romans at one point could not recruit enough troops from within the empire.