TL;DR: There's many studies which point to better mental health outcomes as a result of treatment, such as hormones and surgery, of transgender individuals.
It's also important to consider that there's a range of gender identities and a range of treatments which are tailored to the individual. There's a lot of factors to consider.
WPATH Standards of Care
The best place to start would be the WPATH Standards of Care, available from www.wpath.org. I'll quote from this:
Although Harry Benjamin already acknowledged a spectrum of gender nonconformity (Benjamin, 1966), the initial clinical approach largely focused on identifying who was an appropriate candidate for sex reassignment to facilitate a physical change from male to female or female to male as completely as possible (e.g., Green & Fleming, 1990; Hastings, 1974). This approach was extensively evaluated and proved to be highly effective. Satisfaction rates across studies ranged from 87% of MtF patients to 97% of FtM patients (Green & Fleming, 1990), and regrets were extremely rare (1-1.5% of MtF patients and <1% of FtM patients; Pfäfflin, 1993). Indeed, hormone therapy and surgery have been found to be medically necessary to alleviate gender dysphoria in many people (American Medical Association, 2008; Anton, 2009; The World Professional Association for Transgender Health, 2008).
It's also worth noting that there's a range of treatments which are individualized:
Treatment is individualized: What helps one person alleviate gender dysphoria might be very different from what helps another person. This process may or may not involve a change in gender expression or body modifications. Medical treatment options include, for example, feminization or masculinization of the body through hormone therapy and/or surgery, which are effective in alleviating gender dysphoria and are medically necessary for many people. Gender identities and expressions are diverse, and hormones and surgery are just two of many options available to assist people with achieving comfort with self and identity.
Example research papers
There's many research papers which compare mental health of transgender people before and after procedures. For example:
In a sample of 359 gender-dysphoric persons who completed "several psychometric measures":
... during [cross-sex hormonal treatment], [gender-dysphoric persons (GDs)] reported a significant reduction of general psychopathology, depressive symptoms, and subjective GD, whereas social and legal indicators of GD showed a significant increase across time ...
Fisher et al., Cross-Sex Hormone Treatment and Psychobiological Changes in Transsexual Persons: Two-Year Follow-Up Data, The Journal of Clinical Endocrinology & Metabolism, 2016.
In a sample of 61 participants who were surveyed:
The present study suggests a positive effect of hormone therapy on transsexuals' QoL after accounting for confounding factors. These results will be useful for healthcare providers of transgender persons but should be confirmed with larger samples using a prospective study design.
Gorin-Lazard et al., Is Hormonal Therapy Associated with Better Quality of Life in Transsexuals? A Cross‐Sectional Study, The Journal of Sexual Medicine, 2012.
Here, QoL = Quality of Life.
In a meta review for transgender men:
Multiple studies have shown the positive effect of gender affirming hormonal therapy and gender affirming surgery on quality of life of trans persons and several studies describe an increase in their psychological wellbeing. In addition, satisfaction rates after gender affirming surgery are high and surgery is rarely regretted. However, as only one study has addressed cost-effectiveness of gender affirming treatment in trans men, further research is necessary.
Defreyne et al., Healthcare costs and quality of life outcomes following gender affirming surgery in trans men: a review, Expert Review of Pharmacoeconomics & Outcomes Research, 2017.
There's many more I haven't listed; they can be found by searching PubMed.
Also note that Johns Hopkins have subsequently reaffirmed their stance:
We have committed to and will soon begin providing gender-affirming surgery as another important element of our overall care program, reflecting careful consideration over the past year of best practices and the appropriate provision of care for transgender individuals.
Johns Hopkins Medicine's Commitment to the LGBT Community, October 7, 2016.