That is what I thought you were asking for - and it isn't something that should be done in SQL. A totals line is part of a report and should be done in the presentation layer.
However - you could use Grouping Sets to get something close to what you are looking for:
Declare @testTable Table (Fornavn varchar(35), Efternavn varchar(35), Start_dato date, Slut_dato date, Antal_dage int);
Insert Into @testTable (Fornavn, Efternavn, Start_dato, Slut_dato, Antal_dage)
Values ('Susanne', 'Olsen', '2020-07-01', '2020-07-08', 8)
, ('Susanne', 'Olsen', '2020-10-12', '2020-10-19', 8)
, ('Susanne', 'Olsen', '2021-01-22', '2021-01-22', 1)
, ('Susanne', 'Jackson', '2020-07-03', '2020-07-08', 6)
, ('Susanne', 'Jackson', '2020-10-14', '2020-10-18', 5)
, ('Susanne', 'Jackson', '2021-01-20', '2021-01-21', 2);
Select tt.Fornavn
, tt.Efternavn
, tt.Start_dato
, tt.Slut_dato
, sum(tt.Antal_dage) As Antal_dage
, Antal_dage_2 = sum(datediff(day, tt.Start_dato, tt.Slut_dato) + 1)
From @testTable tt
Where tt.Fornavn = 'Susanne'
And tt.Start_dato > '2020-01-24'
Group By Grouping Sets (
(tt.Fornavn, tt.Efternavn)
, (tt.Fornavn, tt.Efternavn, tt.Start_dato, tt.Slut_dato)
);
I added an additional person to the test table to show the results when you have more than a single person selected. You will get the first name (Fornavn) and last name (Efternavn) included in the results but I think that is much more clear than having blank values.
I also added a new column that calculates the number of days from the start date (Start_dato) and end date (Slut_dato) to show how you can get that value without having to pre-calculate the number of days (Antal_dage).
Note: in the grouping sets - the first set (tt.Fornavn, tt.Efternavn) gives you the totals by name and the second set ((tt.Fornavn, tt.Efternavn, tt.Start_dato, tt.Slut_dato) is required to include Start_dato and Slut_dato in the grouping.