Despite the question being edited multiple times, I don't feel the principal matter of the question is clear. How to assess suicidal tendencies in an 11-year-old? Are we assuming this is indeed expression of suicidal ideation and so OP is interested in knowing what intervention strategies are being used with suicidal children? Is OP looking for socially supportive answers? Are answers comparing the efficacy of various therapeutic strategies in suicidal children expected? And, how does whichever of these intentions jive with OP's expressed intent of seeking in-person professional help, which IMO should be given precedence in clinical matters, such as the suicide risk.
Suicide risk assessment
In my personal opinion, this is firmly outside the scope of this site.
Most risk assessment measures for “youth” are intended for use with adolescents, and the nature of children’s developmental functioning presents particular challenges for accurate assessment. 
[...] assessment of childhood suicide [is] one of the most difficult of diagnostic tasks. 
Entire books could be written on suicidal risk assessment in children. There are various scales, with various precision, various internal consistency, various reliability or interchangeablity - but all are to be administered by a professional in the field. In the hands of a layperson, assessment strategies may provide some mental scaffolding for an adult to be able to tackle the concept of possible suicidal ideation in children, but in concrete terms, actual professionals need to be consulted. They will also be able to help when deciding how to frame a therapy visit to a child with history of resenting therapy sessions.
Intervention strategies with suicidal children
This topic is also encompassing enough to have books and books written on it. There are a great many therapeutical schools and psychiatric approaches; OP has expressed only 1 criterion - reluctance to employ pharmaceutical approaches. Upon closer look, it seems to only be with regards to anti-depressants. (Suicidal ideation and tendencies can of course be closely related to child depression, but as we cannot even remotely begin to assess depression in OP's daughter, we are considering all possibilities - an example in suicial children is hallucinations - is OP also opposed to anti-psychotics, or is asking the community to evaluate whether other psychiatric drugs have enough research-based factual support for him to agree to medicating his child?)
As the rules say,
And some subjects are considered off-topic because they're not directly related to parenting,
specific medical issues (we're not doctors!)
I cannot think of any possible answer here other than "take your daughter's suicide threats seriously (keep her under supervision, remove dangerous objects so that she can't get to them), consult professionals to make a plan on how to address this, follow through. Seek a second opinion if you feel your concern is not taken into account."
Comparative efficacy of different approaches
This is not a parenting topic as the approaches themselves are not usually done by parents. I would recommend against employing parenting strategies
to resolve suicidal ideation in a child. Unless the parent is causally involved somehow (the parent also suffers from suicidal ideation/depression, or an idealised view of death has been presented to the child, or the parent engages in destructive behaviors that are causing severe instability and distress in the child, etc) it is a question of the child's health, separate from the topic of parenting.
OP already seeking in-person professional help
IMO this is a case of:
avoid asking subjective questions where ... your answer is provided along with the question, and you expect more answers
If there is to be anything added within the scope of this site, it is only going to be an answer such as I outlined under the Social Support heading.
Depression and suicide in children is a difficult topic
It is bound to provoke an emotional response (also provoked one in me where I saw the topic in the queue, knew OP was asking for an exceptionally wide overview, but the topic of child suicide elicited a strong gut reaction to the point I emotionally perceived it to be so important, site rules could/should be disregarded). However, I think that no assessment should be done via Internet in such matters, not even weighing in saying "it looks like your child may just be trying to manipulate you". Even if it looks like it. Because it is desirable to err on the side of caution, I think disallowing such questions is the way to go, hand in hand with expressing an umbrella policy of taking children suicide talk/threats seriously by default.
Perhaps a short resource could be compiled for parents in shock after discovering their child talking about/planning suicide, pointing to mental health/suicide prevention websites, hotlines, professionals, or registers thereof, along with a basic instruction set of 1. keep calm 2. keep an eye on the child 3. seek help 4. keep your wits about you. Then again, such articles are most likely not difficult to find elsewhere on the internet (I have come across several after a cursory search.)
: Barrio, Casey A. Assessing Suicide Risk in Children: Guidelines for Developmentally Appropriate Interviewing. Journal of Mental Health Counseling, v29/n1/p50-66/Jan 2007
: Wise, Amy J.; Spengler, Paul M. Suicide in Children Younger than Age Fourteen: Clinical Judgment and Assessment Issues. Journal of Mental Health Counseling, v19 n4 p318-35 Oct 1997