Mrs V, together with her children, jog or jogs in the park every weekend. Thank you.
To choose between the singular 'jogs' and the plural 'jog', delete the parenthetical clause "together with her children", and consider each verb form.
Mrs V jog in the park
Mrs V jogs in the park
Which one is right? The second one. Mrs V, she of the curiously short surname, is one person, a singular entity.