Why is it so important to the excecutive branch that there is a full Congress? Can action not be taken by the sole remaining Congressman, or is the problem that this Congressman would have way too much power?
No, action cannot be taken by the remaining Congresspeople1
Section 5 Clause 1 of the Constitution says:
Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members, and a Majority of each shall constitute a Quorum to do Business;
There are 100 senators and 435 members of the house - you need 51 senators and 218 members for each house to "do Business". Until at least half the vacancies are filled no business of Congress can be conducted.
When there are casual vacancies in the Senate (e.g. because the Senator got blown up during the State of the Union), the Seventeenth Amendment requires that an election be held to choose a replacement. The amendment also allows states to empower their governors to appoint a replacement until such election can be held. In practice, a majority of states (but by no means all) allow the governor to appoint a replacement, and then hold an election for the replacement at the time of the next general election.
When there are casual vacancies in the House, the legislature of the relevant state has to call an election to fill the vacancy (Section 2 Clause 4).
1 My recollection is that there was more than one survivor and I distinctly remember the Speaker was a woman.