Now, the only way is getting a mod to look at the question is flagging. So, I used custom flag and asked moderators to close the question. I can't use my close vote again because it is already used up a week ago.

This is not entirely correct. Your close vote is actually only temporarily used up. 14 days after it expires you will be able to cast it again. As per Shog9's post about these changes:

Allow re-casting votes that've aged away after 14 days. That is, 14 days (use the same site-configurable value used in #1 here) after your vote has aged away, you're free to cast the same vote again.

The problem you are facing in your particular situation is described in that post as well, and it is one of the reasons why the change was made to enable recasting votes:

Some folks raise a flag when this happens, but moderators are often reluctant to intervene on topics they aren't personally familiar with unless the problem is truly egregious - they signed up to be exception-handlers after all, and a question no one cares about isn't all that exceptional. Others lean on the Very Low Quality flag to give questions a second shot at review - but strictly-speaking, this is an abuse of VLQ and will probably get harder as we continue to develop review.

Essentially, you can recast your close vote infinitely until it is actually effective. If a question fails to get closed after you repeatedly send it to the Close Queue then one of two conclusions is likely:

1. The question is not actually closeworthy.
2. A small minority of users are hogging the queue and kicking out the question before anyone else can see it and vote to close.

In the first instance, asking moderators to close the question would probably not work. If the community doesn't want a question closed (as evidenced from the repeated failures in the queue) then it is not a moderator's job to override that (unless there is a specific policy being explicitly violated).

In the second instance, the particular question is merely a symptom of a larger issue with the site not functioning properly. If you address that issue then the issue of particular questions not getting closed will end up resolving on its own.

So in sum, if your close vote fails you should keep trying. It may be frustrating to have to wait a few weeks between each recast, but on the bright site yo can keep track of the votes you need to recast in the Closure tab in the Votes section of your Activity page.

Now, the only way is getting a mod to look at the question is flagging. So, I used custom flag and asked moderators to close the question. I can't use my close vote again because it is already used up a week ago.

This is not entirely correct. Your close vote is actually only temporarily used up. 14 days after it expires you will be able to cast it again. As per Shog9's post about these changes:

Allow re-casting votes that've aged away after 14 days. That is, 14 days (use the same site-configurable value used in #1 here) after your vote has aged away, you're free to cast the same vote again.

The problem you are facing in your particular situation is described in that post as well, and it is one of the reasons why the change was made to enable recasting votes:

Some folks raise a flag when this happens, but moderators are often reluctant to intervene on topics they aren't personally familiar with unless the problem is truly egregious - they signed up to be exception-handlers after all, and a question no one cares about isn't all that exceptional. Others lean on the Very Low Quality flag to give questions a second shot at review - but strictly-speaking, this is an abuse of VLQ and will probably get harder as we continue to develop review.

Essentially, you can recast your close vote infinitely until it is actually effective. If a question fails to get closed after you repeatedly send it to the Close Queue then one of two conclusions is likely:

1. The question is not actually closeworthy.
2. A small minority of users are hogging the queue and kicking out the question before anyone else can see it and vote to close.

In the first instance, asking moderators to close the question would probably not work. If the community doesn't want a question closed (as evidenced from the repeated failures in the queue) then it is not a moderator's job to override that (unless there is a specific policy being explicitly violated).

In the second instance, the particular question is merely a symptom of a larger issue with the site not functioning properly. If you address that issue then the issue of particular questions not getting closed will end up resolving on its own.

So in sum, if your close vote fails you should keep trying. It may be frustrating to have to wait a few weeks between each recast, but on the bright site you can keep track of the votes you need to recast in the Closure tab in the Votes section of your Activity page.