The core challenge here is to minimize concerns that your "self-employment" consisted of playing video games and hanging out with friends talking about an idea that never went anywhere because you weren't working or lacked dedication. Employers actively avoid hiring someone that claims self-employment in the absence of real work.
Also, while the idea of "even though you failed, you learned valuable lessons" is great, it is not true. You might be an idiot, lazy, disorganized, unprofessional, etc. which is why you failed. This is a valid problem for an unverifiable "job" and you are right to be concerned.
So you need to demonstrate:
- You no longer want to be self-employed. This has to be very clear in your resume, or else you risk the appearance of trying to moonlight to keep your current idea alive; you will only be half-heartedly dedicated to your employer while really trying to keep your start-up alive.
EDIT:
On your resume, to demonstrate a firm commitment to employment, state: business liquidation, filing of a final IRS return, legal closure of the business with the state of incorporation, a link to a webpage that states you are no longer supporting the product, etc. If not this, you can emphasize financial or personal reasons that you are seeking employment (like the discovery of excessive development or operational costs, unanticipated time spent on non-technical aspects of self-employment, a clear admission of defeat to the competition, publicized events that severely damaged your product/vision/operations, etc.).
The key element here is that if your resume does not communicate a clear and urgent need to abandon self-employment, potential employers may not be convinced that you will be committed to employment: i.e. subordinate to a manager or boss. Many people leave employment to "do their own thing" so you must be convincing that you are motivated to be compliant and dedicated as an employee.
END EDIT.
Concrete evidence of lessons learned. If you spent $500 on a Facebook ad campaign, say that explain what you learned from it. If you hired a designer to create UI and realized that it would take too long to produce, you need to mention details about that. No employer wants a failure, but more so they don't want you walking in to a job making more unrealistic estimates or have unreasonable expectations about what you can accomplish during your employment with them.
On the job learning. Provide links to websites that show current skills (your company website, github, stackoverflow...). Your company has no credibility, so no one can know if you have up to date skills or none at all. Anything to help them see that you actively learned is important.
Other people respect your work. Get references from your UI designers, marketers and anyone else that provided professional advice or services to your startup. Credibility is critical. It's nice that you say how you engaged others, but if your "UI designer" is your sister and she's not a web person, you are right to be worried.
You can complete tasks. Provide links to websites, app stores or any place else that references work that you completed while trying to do your startup. Mention prototypes, focus group results, surveys, etc. This is about showing that you don't "give up" or get distracted. That you will finish what you start with your employer.
That you are professionally active. Include attendance at trade shows, conferences, meet-ups or anywhere else that you had a professional presence. This demonstrates active learning and professional development on your part with outside resources that are verifiable and can be asked about during your interview.
Not all of these are necessary, and you may have other things not listed that will help demonstrate your value to a potential employer.
