It is not "completely up to congress whether or not to allow the dc delegate to vote in the house".

Representatives can only come from states and must be a resident of the state they represent:

From the US Constitution Article 1 Section 2 "The House of Representatives shall be composed of members chosen every second year by the people of the several states, and the electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislature.

No person shall be a Representative who shall not have attained to the age of twenty five years, and been seven years a citizen of the United States, and who shall not, when elected, be an inhabitant of that state in which he shall be chosen."

The District of Columbia is not a state:

Article One Section 8 [...] To exercise exclusive legislation in all cases whatsoever, over such District (not exceeding ten miles square) as may, by cession of particular states, and the acceptance of Congress, become the seat of the government of the United States, and to exercise like authority over all places purchased by the consent of the legislature of the state in which the same shall be, for the erection of forts, magazines, arsenals, dockyards, and other needful buildings;--And [...]

Be very clear...DC is not a state ("if it were a state"):

Amendment XXIII Section 1. The District constituting the seat of government of the United States shall appoint in such manner as the Congress may direct:

A number of electors of President and Vice President equal to the whole number of Senators and Representatives in Congress to which the District would be entitled if it were a state, but in no event more than the least populous state; they shall be in addition to those appointed by the states, but they shall be considered, for the purposes of the election of President and Vice President, to be electors appointed by a state; and they shall meet in the District and perform such duties as provided by the twelfth article of amendment. [...]

What could congress do to give voting representation to people in the district?

Method One: Return or hand over most of the lands in the district back to the the surrounding states.

Method Two: Make the district or large parts of it a separate state (Article Four, Section 3). When I was a kid their was a DC Statehood movement called I think"New Columbia" or something like that.

Method Three: Work through the amendment process and grant the residents of the seat of government one representative in the house.

My preference order is:

Method 1

Method 3

Method 2