It's tricky. If someone is determined to die there's really nothing you can do to stop them. To get help your friend has to engage with the help offered, shite though it often is. It's totally not your responsibility to solve this. However, if someone can identify barriers to her seeking and accepting help, maybe solutions to those barriers can also be identified. Sometimes there are genuine reasons why someone isn't engaging with the help offered, sometimes it's because the help is totally unsuitable for them and they can't engage with it. The MH services don't always get it right.
This is going to sound awful but some people just like to moan and moan and moan and do nothing to change their situation. Or maybe their situation genuinely can't be changed at all, but IME that's rare, it's usually that they don't want/like the options for change that are available to them.
If you've reached the end of your capacity for listening to a suicidal person day after day, it's ok for you to stop doing it. I know you haven't asked about that but I'm throwing it out there anyway, because stopping being a listening ear to someone who's troubles are draining you down and seriously negatively impacting on your own life often isn't something most people feel is ok for them to do. They instead feel misplaced responsibility or guilt. So I'm just pointing out that you're not responsible for your friend's life.
Sometimes by endless listening and sympathising we're actually enabling, albeit accidentally. The person has someone to moan to which relieves the load enough that they're able to carry on as they are, so they don't ever attempt to do anything to change the situation or their life for the better. Leaving both parties caught in a vicious circle. Without the enabler the person would be forced to take action. The only problem, especially in a suicidal person, is what that action may be. That's totally on them though. And there are various MH charities and helplines people can call. Although that isn't the same as talking to someone who knows their history and who they have a rapport with. I'm wondering if instead of calling you with her suicidal feelings, your friend whould be better served by calling her MH professional (if she has one)?
You can call the police to do a welfare check if someone tells you they're suicidal. That won't help if she tells them she's fine but it's something you can do. You can write to their GP or MH professional, if you know who that is, telling them of the situation with the daily phone calls etc. You won't get a response because of patient confidentially but you'll be making them aware of the truth of the situation. People can seek help privately if they don't want to use the NHS, either paying themselves or perhaps accessing some form of help through an employee's health insurance package as part of their salary benefits or student's in-house university provision of MH support. I'm not sure exactly what help may be available from charities or helplines, it's possible your friend is too unwell for this to be of much use right now, but could be worth finding out.
Beyond that there's really not much you can do. Suicidal person only gets sectioned if they have immediate plan to carry it out.
Ultimately you can't force a person to accept help though, as much as you might want to.