Learning needs analysis tends to be seen as a bit of a dark art, and the truth is that there is no standard way of conducting one since it often depends on a range of factors.
What experienced L&D/OD practitioners tend to do is rely on a great deal of tacit knowledge to come up with the right approach. However, there are some activities that you can nearly always undertake in order to get useful information to inform your L&D strategy and I thought I'd outline some of these.
It's important to note that all of these will highlight areas where there are performance needs. Whether the solution is some form of learning intervention or a change to some other part of the workplace performance ecology is the subject for a future post.
Interview a cross-section of managers (including senior managers)
It's important to note that you should not be asking the managers what L&D interventions they think their staff need, but rather ask them about what they need to achieve in the coming year/quarter, what the key deliverables of their business plans are, what challenges they think they and their teams/departments might face, and what they personally have found to be the most challenging aspects of their jobs in the last three months. When you interview them, you should solicit real life examples rather than vague generalizations. Recording the meeting and making thorough notes afterwards will allow you to look for any trends across the business or in particular areas. 
Survey a cross-section of all your (target) employees
To get some quantitative data, a survey is a very easy way to get useful information to feed into your analysis. As with the interviews, it's critical that you don't ask questions like, "How useful would you find a course on presentation skills?" or, "What training would help you to achieve your goals?" because you won't get at what the genuine needs are. Rather ask about what's challenging about their jobs now, and what they believe future challenges will be based on the future strategy. SurveyMonkey is a brilliant tool for collecting and analyzing this type of information.
Analyse the development plans of a sample of your target population
You don't need to review all 1500 development plans, just a significant enough a sample to identify the key development requirements across the employee base. You can do this by textual analysis or even a simple tallying process. Again, look for trends and discount any plans that simply have 'communication skills course'. You're looking for what the real needs are, not what managers and employees think is the right solution. You're the L&D expert.
Review search terms on your LMS and/or intranet
Your IT department will definitely be able to get this information for you and it's a rich source of information. Take a look at this video where the Head of Google Learning Labs describes what happened when they conducted a similar exercise (skip to around 15 minutes in). He and his team did an extensive traditional TNA and compared the top search terms on their learning portal against the top learning needs identified in their TNA - they matched exactly! I wouldn't want to rely on the results of search terms on their own, but they're definitely worth reviewing to see what they can tell you.
Review business plans for the key areas in your organization
You'll get this information from the manager interviews, but it's worth reviewing them to make sure that you (or they) haven't missed anything.
Review key business metrics and trends
Understanding where the organization is failing to hit its key metrics can lead you to some hidden needs. If you work for a large organization, the people responsible for producing the organization's annual report usually have this information to hand. Otherwise, you may need to get it from key departments individually.
The kind of danger signals you're looking for are things like:
- Increased customer complaints
- Increased absenteeism
- Lower sales
- Increased product defect rates
- Increased staff turnover
- Deteriorating financial indicators
You don't have to do all of these if you're pushed for time, but a training/learning/performance needs analysis should be thorough. It's the basis of what you spend your budget on and you want to make sure that your budget is spent as effectively as possible.
 Alternatively, you may want to try running a few focus groups. This has the advantage of getting greater numbers, but there's the danger you run into groupthink and, ultimately, I think interviewing is a better approach overall.