Sounds like you got a knockoff Vs a real unit as the outer covering is silicon, unlike the knockoffs which use vinyl or PVC which over time gets brittle.
Time for a new charger, the cord is not really serviceable ;-{
[https://www.apple.com/us/search/MagSafe+Power+Adapter?src=globalnav|MagSafe Power Adapters]
● [https://lifehacker.com/dont-replace-your-macbook-charger-with-a-cheap-knockoff-1825605569|Don't Replace Your MacBook Charger With a Cheap Knockoff]
● [https://beetstech.com/blog/oem-magsafe-chargers-vs-counterfeits-teardown-for-truth|OEM MagSafe Chargers vs Cheap Imposters: Teardown for Truth]
● [http://www.righto.com/2016/09/why-you-shouldnt-use-cheap-macbook.html|Lacking safety features, cheap MacBook chargers create big sparks]
● [https://www.youtube.com/watch?v=DrxAuhrrLU4|FAKE Magsafe MacBook Chargers on eBay - Watch before buying a Mac Charger]