According to this legal analysis, the answer is no.
My review of Senate precedent suggests the answer is no. A CRS report notes that “[p]revailing practice is for state governors to fill Senate vacancies by appointment, with the appointee serving until a special election has been held, at which time the appointment expires immediately.” This practice is in accordance with the Seventeenth Amendment, which provides that “the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.”
The Massachusetts law, passed in September to authorize Kirk’s appointment, provides that an appointed Senator shall serve “until the election and qualification of the person duly elected to fill the vacancy.” This would seem to support the position that Kirk can continue to serve after the special election is held. However, the Senate has previously found that substantially similar state laws cannot extend the term of an appointed Senator beyond the date of the special election.
On May 7, 1937, George Berry was given a temporary appointment as U.S. Senator from Tennessee to fill a vacancy created by the death of Senator Nathan Bachman. On November 8, 1938, a special election was held to fill the seat. In accordance with Tennessee’s normal practice, it took several weeks before the votes were counted and a winner was not certified until January 3, 1939. The applicable Tennessee law provided that a temporary Senator “shall hold office until his successor is elected at the next biennial election and qualifies.” Based on this law, Berry claimed that he was entitled to hold office and be paid until his successor was certified and/or actually seated by the Senate.
Berry’s claim was referred to a subcommittee of the Senate Judiciary Committee, which held a hearing and considered legal arguments on the matter. A legal analysis prepared for the subcommittee found that “in view of [Seventeenth Amendment’s] purpose of providing for representation in the Senate by persons elected by popular vote both for full terms and for unexpired terms it seems reasonable to assume that no temporary appointment was to be authorized except for the intervening period between the creation of a vacancy and the day when the people by their votes actually elect a successor, or, in other words, until they elect a person to fill the vacancy.”
In addition to the text and purpose of the Seventeenth Amendment, the analysis relied on various Senate precedents, including an October 15, 1918 ruling by Vice President Marshall, who found that the phraseology of the amendment was “radically different” than that of various state laws that permitted appointees to serve until their successors were “elected and qualified.” Marshall concluded that regardless of the fact that Senators-elect must “run the gamit of executive, administrative, judicial and senatorial investigation before they are entitled to qualify and take their seats as Members of the United States Senate,” the terms of their appointed predecessors nonetheless expire on the day of election. While the Vice President noted that “[e]quitably, it would seem that the present incumbents ought to be permitted to hold until their successors elected on the 5th of November have been sworn in as Senators,  such . . . is not the law.”
The Senate subcommittee and committee concluded, based on its hearing and review, that “the term of service of a Senator appointed to fill a vacancy in an unexpired term ends on the day when his successor is elected by the people.” 1939 Congressional Record, p. 998. There was evidently no controversy among either the subcommittee or full committee regarding this legal conclusion, and the committee then presented a resolution to the Senate for adoption, expressing the view that Berry’s term of service expired on November 8, 1938, the date of the special election. As Senator Connally, a member of the subcommittee, explained to the Senate, the fact that the Tennessee statute purported to extend Berry’s term until the qualification of his successor was of no force because the statute was “plainly in conflict with the provisions of the seventeenth amendment.” Accordingly, the Senate adopted the proposed resolution without dissent. 1939 Congressional Record, p. 1058.
Based on this authority, it would appear that a valid point of order could be raised as to Senator Kirk’s participation in Senate proceedings after January 19, 2010.