Three weeks into her term, Mayor Katie Wilson asked Seattle’s head of housing for her resignation. Four months later, she’s yet to announce her pick for a permanent replacement. The struggling affordable housing sector is anxiously waiting to see how Wilson’s chosen successor to the job will tackle the existential problems it faces — financial problems that have led to sold buildings and cut services . This person will also influence how the new administration will spend the hundreds of millions of dollars Seattle collects for housing. The job is still posted on public-facing job sites like Indeed, and several providers say they’ve heard little. By contrast, Wilson’s predecessor, Bruce Harrell, named a new director in January of his first year. The mayor is hopeful to have a nominee by July, said spokesperson Sage Wilson.…