The Eastern courts reaction to the fall of the Western Empire (if that's what you're referring to and not the Sack of Rome in 410) could probably be best described as a regretful shrug.

The collapse of the Western Roman Empire wasn't a sudden, catastrophic event. It was a very long, gradual process that seems obvious to us today, with hindsight, but wasn't as obvious to people alive at the time.

From the Eastern Empire's perspective the West had already been lost by the time of Odoacer's coup, the last Emperor recognized by the East was Julius Nepos (who prior to this had been sent to Italy by the Eastern Empire). Romulus Augustus was never recognized by Constantinople and was seen as illegitimate.

Infact, out of all the Western Emperors following Majorian, only two were ever recognized by Constantinople at all (that being, Anthemius and Julius Nepos), and both of these originally came from the east to try and limit Ricimer's influence. It is absolutely worth emphasizing that the Western Empire was an "Empire" in name only following Majorian's assassination. Ricimer had essentially been the one running the show in Italy for decades, and the Emperor was mainly a loyal figurehead who rubberstamped everything Ricimer did. If any Emperor tried to cross Ricimer, they would be killed, such as Anthemius. There was even a full year (466) where there was no Emperor at all, Ricimier simply didn't see the point in appointing one until the Eastern Empire forced their candidate.

Odoacer proclaiming himself "King of Italy" and abolishing the position of Emperor was really nothing more than a confirmation of what had been the reality on the ground for decades. Ricimer had himself essentially been "King of Italy" since the 460's and the Western Emperors were merely for show. The Eastern Empire was very aware of this, and as such, Odoacer's coup was more or less met with a shrug.

That's not to say they didn't try to prevent it though. When Odoacer abolished the office of Emperor, he sent the Roman Senate to Constantinople so they could give the Eastern Emperor (Zeno) his proposal. His proposal was simple, the west no longer had a need for their own Emperor, and the Empire was going to be reunited under one Emperor once more, Odoacer would rule Italy as a Patrician under Zeno's suzerainity.

Zeno, suprisingly, refused this offer and told the Senate that they needed to recognize Julius Nepos. Julius Nepos (Emperor prior to Romulus Augustus) no longer ruled Italy, but he was still Emperor-in-Exile in Dalmatia. There wasn't much Odoacer could do, so he begrudgingly recognized Nepos and minted new coins with his name on them. Nepos never did return to Italy, possibly because he feared for his life, and possibly because he may have been trying to gain support from the Eastern Empire to retake Italy.

Nepos' ambitions came to an end however in 480, when he was assassinated by his guards, in a conspiracy which may have been orchestrated by Odoacer (or perhaps former Emperor Glycerius, who also lived in Dalmatia as a bishop), although we'll never quite know for sure.

Following Nepos' death, Zeno decided to finally revert to Odoacer's original offer, and "ruled" Italy in-name only. Zeno however, gradually grew more distrustful of Odoacer throughout the 480's, and began seeing him as a rival. The Eastern Empire was also struggling with raids from the tribe of Ostrogoths settled in Illyricum. Zeno, killing two birds with one stone, decided he would release Theoderic, an Ostrogothic Prince who was held hostage and educated in Constantinople, to lead his tribe in an invasion of Italy. The Ostrogoths were to depose Odoacer, and rule Italy under Zeno, thereby Zeno would rid himself of both Odoacer and the Ostrogoths.

The Ostrogothic invasion was a success, they swiftly conquered Italy in a few years and Theoderic was able to kill Odoacer during a faux-celebration banquet. The Ostrogoths would continue to rule Italy, in the name of the Eastern Emperor, until Justinian's invasion in the 530's-550's.

In short, the Eastern Empire's reaction was initially not all too dramatic. They had seen the West as essentially already lost, and Odoacer's coup merely confirmed that fact. However they did make several attempts in the years afterward to undermine Odoacer and cement Roman authority in Italy. Either by propping up Julius Nepos in Dalmatia, or releasing Theoderic to conquer Italy.

Justinian's reconquest was in many ways a continuation of this policy, although much more dramatic. Instead of vying for influence in Italy, Justinian decided to plainly invade and annex it instead, which ended up being more of a net loss for the Empire in the long term