It depends on what definition of engagement you are relying on. Usually, work engagement is studied based on the work by Bakker and Schaufeli and measured using the Utrecht Work Engagement Scale (UWES) - http://www.wilmarschaufeli.nl/publications/Schaufeli/251.pdf

Good luck!

You can use both a quantitative and qualitative methods. The qualitative method will provide the insights while the quantitative will support it statistically. See the qualitative study below

URL: http://dx.doi.org/10.5539/ijbm.v11n12p148

Engaging Staff in the Workplace, available at https://www.researchgate.net/publication/254582894_Engaging_Staff_in_the_Workplace, defines engagement and its benefits, underscores that surveys can help gauge commitment, lists common barriers to workforce motivation, and isolates 12 elements of great managing. With such comprehensive coverage, that paper ought to provide sufficient food for thought to refine the research proposal.

Article Engaging Staff in the Workplace

The Gallup 12 question survey is an industry standard I think - I assume as it has been around for a while that it has internal validity etc. 

https://q12.gallup.com/

Overall, the UWES is the most established scale to measure employee engagement. However, for the sake of enhancing your research design and reduce common method bias because of (I suppose) single source ratings, you could also rely on proxies such as employee absences, turnover or productivity to assess whether enployees are engaged in their work. Best regards

Think, too, about whether you plan to work with a single survey (which provides only a snapshot in time- current factors affect engagement,  so that must be considered), or whether you are going to look into tr3bds and patterns within an organization over considerable time.)

You may want to investigate both surveys that pertain to organizations in general, as well as examine the differences between instruments designed for different sectors.

As mentioned above, one of the most frequently used instruments in academic employee engagement research is the UWES (17-item version). It is based on the engagement definition by Schaufeli et al.

However, the other frequently used definition of employee engagement is the one by Kahn. In this case you might consider using the Job Engagement Scale (JES) by Rich et al.

Both instruments measure pretty much the same 3 dimensions of employee engagement, and it might be interesting to compare the results.

As already expressed by some senior colleagues, it is prudent that you first decide the role work engagement will play in your study: (a) as an independent variable, (b) moderator or mediator, and (c) dependent variable. Having settled on this, you may utilize a longitudinal or a diary methodology - either weekly or daily as work engagement appears to fluctuate daily. A lot of cross-sectional evidence on work engagement exist, therefore, a longitudinal or diary approach will provide a solid scientific contribution. You may also consider engagement at the team level.

Wefald et al (2011) have compared three work engagement measures, so maybe their article will help:

A Comparison of Three Job Engagement Measures: Examining their Factorial and Criterion-Related Validity

http://onlinelibrary.wiley.com/doi/10.1111/j.1758-0854.2011.01059.x/abstract;jsessionid=66ABF1D67449B4B94E5073862BBC46F1.f04t03

Witemeyer et al., (2013) state that consulting firms such as Hay Group, Valtera Corporation, Pricewaterhouse Coopers, Silk Road and Gallup Organization developed instruments to measure employee engagement.   Witemeyer et al.,(2013) state that the Utrecht Work Engagement Scale (UWES) is a popular scholarly instruments to measure employee engagement.

If you are developing a new instrument to measure the construct of employee engagement please refer the article titled “Towards a Conceptualization and an Operationalization of the Construct of Employee Engagement” to get an idea. 

Reference

Witemeyer, H., Ellen, P. and Straub, D., 2013, September. Validating a Practice-Informed Definition of Employee Engagement. In Third Annual International Conference on Engaged Management Scholarship, Atlanta, Georgia.

Iddagoda, A., Opatha, H. H. D. N. P., & Gunawardana, K. D. (2016). Towards a Conceptualization and an Operationalization of the Construct of Employee Engagement. International Business Research, 9(2). http://dx.doi.org/10.5539/ibr.v9n2p85.