The rest of this question:
And what was the process of the cleansing of the temple done? How long did it take?
There is no record in Scripture, that I can recall, of a priest dying in the temple or tabernacle. There are no specific instructions for cleansing of the temple in the event of a priest dying in the temple. The more general instructions for cleansing after death would most likely have been applied. Some of these instructions can be found in the following verses:
With reference to items on which dead swarming things fell – Leviticus 11:32:
And anything on which any of them falls when they are dead shall be unclean, whether it is an article of wood or a garment or a skin or a sack, any article that is used for any purpose. It must be put into water, and it shall be unclean until the evening; then it shall be clean.
Coming in contact with the dead body of a person required a much longer cleansing period. Instead of being unclean for a day or part of a day, those who contacted a dead person were unclean for 7 days and required washing on the 3rd and 7th day. For example, when someone died in a tent – Numbers 19:11-14:
“Whoever touches the dead body of any person shall be unclean seven days. (12) He shall cleanse himself with the water on the third day and on the seventh day, and so be clean. But if he does not cleanse himself on the third day and on the seventh day, he will not become clean. (13) Whoever touches a dead person, the body of anyone who has died, and does not cleanse himself, defiles the tabernacle of the LORD, and that person shall be cut off from Israel; because the water for impurity was not thrown on him, he shall be unclean. His uncleanness is still on him. (14) “This is the law when someone dies in a tent: everyone who comes into the tent and everyone who is in the tent shall be unclean seven days.
Although we have no record of cleansing the temple as a result of the death of a priest, we have the account of the cleansing of the temple and its furniture and utensils from years of accumulated filth in the days of king Hezekiah. 2 Chronicles 29:15-19, tells us:
They gathered their brothers and consecrated themselves and went in as the king had commanded, by the words of the LORD, to cleanse the house of the LORD. (16) The priests went into the inner part of the house of the LORD to cleanse it, and they brought out all the uncleanness that they found in the temple of the LORD into the court of the house of the LORD. And the Levites took it and carried it out to the brook Kidron. (17) They began to consecrate on the first day of the first month, and on the eighth day of the month they came to the vestibule of the LORD. Then for eight days they consecrated the house of the LORD, and on the sixteenth day of the first month they finished. (18) Then they went in to Hezekiah the king and said, “We have cleansed all the house of the LORD, the altar of burnt offering and all its utensils, and the table for the showbread and all its utensils. (19) All the utensils that King Ahaz discarded in his reign when he was faithless, we have made ready and consecrated, and behold, they are before the altar of the LORD.”
We see from the verses above that the cleansing process took no more than sixteen days.