I wasn't quite sure what you wanted, so thought I'd look for other papers around the same subject; they might be of help? However, you might be familiar with these already:
Snapp, S. D., Hoenig, J. M., Fields, A., & Russell, S. T. (2015). Messy, butch, and queer: LGBTQ youth and the school-to-prison pipeline. Journal of Adolescent Research, 30(1), 57-82.
Article Messy, Butch, and Queer
Down & Out: Social Exclusion of LGBTQ Young People and Homelessness.
Report of the study session held by IGLYO in co-operation with the European Youth Centre Budapest of the Council of Europe and with support from FEANTSA
European Youth Centre Budapest / Strasbourg 26-30 October 2015
https://rm.coe.int/1680707d24
Gwadz, M., Freeman, R. M., Kutnick, A. H., Silverman, E., Ritchie, A. S., Cleland, C. M., ... & Bolas, J. (2018). Do Programs for runaway and homeless Youth Work? a Qualitative exploration From the Perspectives of Youth clients in Diverse settings. Frontiers in public health, 6, 112.
Article Do Programs for Runaway and Homeless Youth Work? A Qualitati...
Abramovich, A. (2016). Preventing, reducing and ending LGBTQ2S youth homelessness: The need for targeted strategies. Social Inclusion, 4(4), 86-96.
Article Preventing, Reducing and Ending LGBTQ2S Youth Homelessness: ...
The U.S. Department of Housing and Urban Development OFFICE OF COMMUNITY PLANNING AND DEVELOPMENT. Review of the LGBTQ Youth Homelessness Prevention Initiative Planning Phase
I think the first step is to identify the research question you want to answer. I understand you want to 'end homelessness among lbgtq youth'. However, this is a goal, not the research question. Thus, what question would you ask and want answered to better understand the problem? Does the literature do a good job of telling WHY lbgtq youth are homeless? Understanding the why is often a good place to start because you can't come up with solutions until you know why people are homeless. But maybe there's another question you would like answered? Plus, you need a solid research question before anyone can give you direct answers about the best methods and how to compute the necessary sample to have sufficient statistical power for that method.