IANAL, yadda yadda and hypothetical implementation details (linkage, etc) could make arguments to the contrary. From GPLv3/Wikipedia:
'A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an "aggregate" if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation's users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.'
In other words, your MIT library is an independent work and does not limit the rights imparted by the other pieces (namely, the GPL pieces). This is somewhat important since you (presumably) don't own the copyright to that MIT work and so your options for relicensing it are limited.
Meanwhile, your code is a direct, derivative work of the GPL code and at a minimum, must be made available under the GPL. You can additionally release under other licenses, provided that those licenses do not conflict with the GPL. A "public domain" non-license might work. And again, you must always offer a GPL option.
I don't know how the AGPL applies here. I would guess that it's pretty much the same except you acquire those above GPL rights just by consuming the output of the code (e.g. visiting the webpage).
'A compilation of a covered work with other separate and independent works, which are not by their nature extensions of the covered work, and which are not combined with it such as to form a larger program, in or on a volume of a storage or distribution medium, is called an "aggregate" if the compilation and its resulting copyright are not used to limit the access or legal rights of the compilation's users beyond what the individual works permit. Inclusion of a covered work in an aggregate does not cause this License to apply to the other parts of the aggregate.'
In other words, your MIT library is an independent work and does not limit the rights imparted by the other pieces (namely, the GPL pieces). This is somewhat important since you (presumably) don't own the copyright to that MIT work and so your options for relicensing it are limited.
Meanwhile, your code is a direct, derivative work of the GPL code and at a minimum, must be made available under the GPL. You can additionally release under other licenses, provided that those licenses do not conflict with the GPL. A "public domain" non-license might work. And again, you must always offer a GPL option.
I don't know how the AGPL applies here. I would guess that it's pretty much the same except you acquire those above GPL rights just by consuming the output of the code (e.g. visiting the webpage).