My commiserations. You were very close to getting the score you needed.
Have a look at my page "What to do if your IELTS result is too low" for advice. It lays out your options from retaking the test to asking for a re-mark.
Looking at your result, it looks possible a re-mark could raise your overall score sufficiently. For example, if your writing mark increased it would get a 6 (writing is marked in whole bands) which would give you a combined total of 25. This would average out at 6.25 and be rounded up to an overall score of 6.5. (See this page on the IELTS site for more about how your overall score is calculated.)
After reading my page, consider whether you feel you wrote well enough to get better than a 5 for your writing. Remember that writing too few words or not adequately addressing the topic will lose marks.
Re-marking Listening and Reading is unlikely to result in increased scores as these modules are marked objectively, unlike Speaking and Writing which are more subjective. You would have to be a really good speaker to increase your Speaking score - a 7 is a good mark.
If you feel you should have received a higher score for writing you might like to consider asking for an "Enquiry on Results" (that's the jargon IELTS uses for a re-mark). Only ask for it on the writing module (or possibly writing and listening) - you don't have to have all modules re-marked. Remember that you must apply for an Enquiry on Results within 6 weeks of the date of your test. To apply for an Enquiry on Results contact the Test Centre where you took the test. It will cost around 2/3 the cost of taking the IELTS test but you will get this fee reimbursed if your score is increased.
Please let us know what you decide to do and, if you decide to ask for a re-mark, let us know how you get on.
Thanks for your email asking how long a it takes to get the result of your re-mark. I have answered it here in this forum, rather than by email, so that we can share the information with others.
It can take up to 10 weeks to receive your result as it will be re-marked by a senior examiner in London. Sometimes it is much quicker. It is best to contact the Test Centre where you took the test as they should be able to give you a more accurate estimate of the time needed.