The sentence is stupid, therefore it's difficult to translate it.
As Irbis explained "ajati" is a childish way to express 'to sleep' "spati". "Mi" is a clitic for "meni" 'to me' and has no specific meaning in this particular sentence, where btw, the comma is missing after "ajaj". This comma separates two clauses. So if I rewrite it to a somehow proper sentence, it would look like this:
Lepo spi, pa pridna bodi! -- Sleep well and be dilligent!
You are asking what the verb is. There are actually two and therefore must be separated by a comma:
- ajaj: an imperative form, singular, of the verb ajati /to sleep/
- bodi: an imperative form, singular, of the verb biti /to be/
pridna is a feminine sing. adjective of the masculine form priden: dilligent
pa ... that's a bit difficult ... it usually means 'but', it can also mean 'and' (specially here in Ljubljana). The meaning can always be visible from the context and dialect.
If you need base forms to search in dictionary, you can try this web site: http://besana.amebis.si/pregibanje/
You just enter any form and you will get all possible base forms (along with all forms for those base forms).
And I don't agree with translation "dilligent" here. This is also possible meaning for "priden" (close to "marljiv" or "prizadeven" in Slovenian), but I think meaning here would be closer to "ubogljiv" ("good").
"pa" has many meaning. In a lot of cases you can just skip it, otherwise you can translate it with "but", "however" or "and" (more colloquial).
And comma is not missing here! If "pa" is in meaning "and" ("vezalno priredje") there is no comma (actually you can distinct meaning this way, if you trust the writer to be aware of this).
When I looked at the web site, its all in Slovanish. Its really nice but there's something I don't understand: you have single, dual and plurial, after that, there's first person, second person, and third person? Who is second person if dual means 2 person? I'm a little bit mix up because there is nothing about masculin or femine form. And the past present and futur???
"na" at the beginning is because "napisati" is perfective verb and "pisati" is imperfective verb ("na" is in this case, typical prefixes to create perfective verbs are: do, iz, z, s, se, na, nad, o ,ob, od, po, pre, pred, pri pro, raz, u, v ,vz, za ... - and there is no simple rule when to use one and it also depends on meaning).
No, for a girl the ending is -la:
Napisala bom pismo.
Pisala bom pismo. (I'll be writting a letter.)