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The difference in lengths of the differential pairs are tremendous, which is quite evident in the screenshot. I just don't see how I am supposed to length-match this. Tips will be appreciated!

enter image description here

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    \$\begingroup\$ Length match does not usually (if ever) mean a perfect match. \$\endgroup\$ – Peter Smith Jun 22 '17 at 13:28
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    \$\begingroup\$ Whats the hole diameter and the width of your traces? \$\endgroup\$ – AntoineLev Jun 22 '17 at 13:29
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    \$\begingroup\$ There is always some tolerance so the lengths never have to match exactly. What your tolerance is depends on the frequencies of the signals, so USB speed. You mention nothing about this. Also, grab a PCB with a USB connector from the junk-box. Observe what they have done there. Profit. \$\endgroup\$ – Bimpelrekkie Jun 22 '17 at 13:30
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    \$\begingroup\$ If the length mismatch is << risetime, then simply execute the layout to match the capacitances. That includes any underlying planes (VDD or GND) and the parallel-plate capacitances. \$\endgroup\$ – analogsystemsrf Jun 22 '17 at 14:11
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    \$\begingroup\$ You can meander the trace in other places to even out the length, but those look like some pretty thick traces. USB should be routed with a 90-ohm differential. Something like a 12-mil width with a 4-mil spacing should give you the 90-ohm differential, and you may be able to route the + signal between the pads. Or switch to surface mount, the USB ports aren't difficult to solder by hand. \$\endgroup\$ – Ron Beyer Jun 22 '17 at 14:15
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I found this: http://suddendocs.samtec.com/catalog_english/usb_a.pdf

You are facing problem with routing because your component doesn't have all pins in-line. The part mentioned above has pins in same line and should get you through.

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